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How to Rationalize a Denominator

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Why might you need to rationalize a denominator? In mathematics the term denominator refers to the bottom number in a fraction. You may be wondering in what circumstances may you need to look at rationalizing a denominator, the bottom number in the fraction which you’re working with.

The answer is that if you ever discover a radical expression in your denominator, you’ll need to get to work on rationalizing your denominator.

Do you need to rationalize your numerator if it contains a radical expression?

If you’re curious about whether you need to rationalize your fraction if you discover a radical expression in the top number of your fraction, which is your numerator, your answer is no. There is absolutely no reason to remove a radical expression from your fraction.

How to rationalize your denominator:

To rationalize the denominator you’ll need to multiply your fraction by either a single term or a set of terms which will be able to remove the radical expression in your denominator, which you’re intent on getting rid of. If you believe that you’d find step by step instructions on removing radial expressions from denominators, simply follow the instructions below.

1. Identify the radical expression

Your first step should be to identify the radical expression in your denominator, which you’ll then work on eliminating from your fraction. What is a common example of a radical expression that you may find in one of your denominators? An example of a radical expression, that you may find in one of your denominators is a square root.

2. Find a suitable number to multiply both your numerator and your denominator with

Once you’ve identified your radical expression, such as a square root, it’s time to try and identify a number which you can successfully multiply your top number and your bottom number of your fraction with.

3. Now multiply the numerator and denominator by the radical which you would like to remove from your fraction

If you happen to be rationalizing a fraction which features a monomial, ensure that you your numerator and your denominator are both multiplied by the same number. As what you’re doing is multiplying your factors by 1.

4. Simplify your fraction

Now that you’ve completed the 3 steps listed above, it’s the right time to simplify your fraction.

How to rationalize a binomial denominator:

If you’re trying to remove a radical expression from a binomial denominator instead of a monomial denominator, simply follow the instructions listed below, in order to discover your answer.

1. Assess the components of your fraction

Start off by assessing the components of your fraction as if your fraction contains a sum of two terms in your denominator which are irrational, you won’t be able to multiply your fraction properly.

2 Multiple the conjugate of your denominator by your fraction

3. Simply your fraction

Rationalize the denominator calculator:

Alternatively, if you’d like to rationalize your fraction is a fraction of the time, it’s well worth using a calculator to solve your equation for you.

Hopefully you now have a clear understanding of how to go about rationalizing denominators in fractions, in order to get rid of radicals such as square roots.

How to Confidently solve two step Equations with Integers

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If you’re interested in being able to solve two step equations without any issues, simply learn how to use the simple, fuss free, two step equation which is outlined below.

How to confidently solve two step equations with integers:

Step one:

Shift all your data that doesn’t have the variable, which you’re looking to the opposite side of your equation.

Step two:

Remember that whatever you do to one side of your equation, you’ll need to do to the opposite to the other side of your equation. As an example if you use a + side on the left side of your equation, you’ll need to place a – side on the right side of your equation.

An example of a simple two step equation:

If you’re still not confident about your ability to use a two step equation in order to be able to solve a simple equation, it may help to work through an example exercise. Let’s start off with a fairly simple equation, that is fairly easy to solve but will allow you to confidently test out your knowledge of both of the two steps which you read about above.

The test equation which we will be working with is 4x+3=7. To find the value of x, make sure to follow the directions which were explained in the previous paragraph.

Step one:

As you know your first step should be to shift all of your data that doesn’t have the variable which you’re looking for, to the opposite side of your equation. In our example you need to subtract three from both sides of your equation. You should have written down 4x+3-3=7-3, which should then give you 4x=4.

Step two:

Now you’ll need to go about diving each side of your formula by 4. The process which you should write down now is 4x/4=4/4 and x=1. If the answer which you worked out is different, simply go back to step one and double check all of the processes which you used, to make sure that you didn’t make a simple error, which you’ll easily be able to correct.

An extra example to work through:

In order to cement the information which you have just learned, it’s a great idea to work through one more example. This time the two step equation which you’ll need to solve is 3x to the power of 3+4=28. To find the value of x, simply work through the simple two step instructions, which you have been learning to follow.

Step one:

Again, begin by isolating the given variable in your equation. You should have written down 3x to the power of 4-4=28-4 and 3x to the power of 4=24.

Step two:

Lastly to find the value of x you’ll need to divide each side by 3. Your notes should now show 3x to the power of 3/3=24/3. X to the power of 3=8. Now write down the cube root for each side and you should get your answer, x=2.

Again, if you didn’t obtain the same answer, simply work through the two step by step instructions listed above again.

What does Corresponding mean in Math?

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If you’re wondering what does corresponding mean in math, simply continue reading to discover a simple easy to understand answer.

What does corresponding mean in math?

Simply put the term corresponding in math refers to objects such as angles which appear in the same exact place, in two similar situations.

What are corresponding angles and how do you go about figuring out the value of corresponding angles?

The mathematical term corresponding angles is commonly used to refer to matching angles. What are corresponding angles? When two straight lines are crossed by a third line, which is known as a traversal line, the angles in each matching corner are referred to as corresponding angles. Corresponding angles are always the same size as each other.
As an example if one angle in a matching corner is 118 degrees, the angle is the corresponding matching corner of the second line will also be 118 degrees.
How to find corresponding angles?
If you label each angle from left to right and top to bottom, you can find corresponding angles by looking at the place of each angle. Angle a and angle e, will be corresponding angles. Angle b and angle f will be matching corresponding angles, angle c and angle g will also be corresponding angles and lastly angle d and angle h will also be even, corresponding angles.
What happens if two lines are crossed by parallel lines?
As a second example, if two straight lines, which do not touch are both crossed by two parallel lines, which each cross one of the two straight lines but don’t intersect each other, you’ll also be able to work out various corresponding angles.
In this example, again assuming that each angle is labelled from a to b, going left to right and top to bottom, angle a and angle e are corresponding angles, angle b and angle h are corresponding angles, angle c and angle f are corresponding angles and angle d and angle h are also corresponding angles.
While it may take a couple of minutes to understand the rules which govern corresponding angles, once you’re able to find out one corresponding angle, you should have no issues figuring out corresponding angles in the future.
Are there any other circumstances in maths in which the term corresponding is frequently used?
Yes, the term corresponding is often used to describe properties of congruent triangles. What is a congruent triangle? A congruent triangle is a common type of triangle where each side is exactly the same length as the other two angles. Congruent triangles also feature three internal angles a right angle and two congruent or equal angles.
As congruent triangles feature equal lines and angles, congruent triangles can also be said to feature corresponding lines and corresponding angles. If you’re ever trying to explain the term corresponding in maths to a friend, it’s a great idea to use this particular example, to illustrate the points which you want to make.
Hopefully you’re now confident about how the term corresponding is used in mathematics and how you can effortlessly figure out corresponding angles.

The key features of Regular Polygons

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If you’re interested in learning about some of the features of regular polygons, simply continue reading to discover valuable information on a wide variety of polygons. Examples of which include a 10 sided polygon and a 100 sided polygon.

What is a 10 sided polygon called?

A 10 sided polygon is most commonly referred to as a decagon, although they are sometimes also referred to as a 10-gon.

What are the key features of a decagon?

A decagon is a regular polygon as it has 10 equal sides. All of a decagon’s internal angles each add up to 144 degrees. As a decagon’s internal angles are less than 180 degrees, a decagon can accurately be described as being a convex polygon.

What is a 9 sided polygon called?

A 9 sided polygon is called a nonogan.

What are the key features of a nonagon?

A nonagon has nine equal sides and as such is also a regular polygon. Just like a decagon a nonagon is also classed as a convex polygon as each of the congruent interior angles of a decagon measures 140 degrees and as mentioned above to be classified as an obtuse polygon a polygon needs to feature interior angles which measure at least 180 degrees.

What is an 8 sided polygon called?

An 8 sided polygon is referred to as an octagon. If you’ve ever watched a UFC, mixed martial arts event you’ll know that each fight takes place within an octagonal shaped ring which features 8 equal sides.

What are some of the key features of an octagon?

An octagon features 8 sides as well as 8 interior angles. Each interior angle inside an octagon measures exactly 135 degrees. An octagon is also a convex polygon.

What is a 7 sided polygon called?

A 7 sided polygon is most commonly referred to as a heptagon. However, a 7 sided polygon is also sometimes referred to as a septagon as “sept” is a latin prefix which is linked to the number 7. Whereas the term heptagon has Greek roots as the prefix “hepta” is connected with the number 7. If you’re curious the “agon” is also Greek and roughly translates to angle. So in Greek the term heptagon translates to 7 angles.

What are some of the key features of a heptagon?

By now you may have guessed that a heptagon has 7 congruent sides and 7 interior angles. Like all of the polygons listed above a heptagon is also a convex polygon as its interior angels each measure 123.57 degrees.

What is a 100 sided polygon called?

While we’ve been discussing polygons which have 10 or less sides, there are polygons which have a far greater number of edges. For starters, a regular polygon which boasts 100 edges is referred to simply as a hectogon. Although you can also accurately call a 100 sided polygon a hectatontogon.

What is a 10000 sided polygon called?

While it can be difficult to picture what a 10000 sided polygon would look like a 10000 sided polygon is referred to as myriagon. Like all of the polygons listed above a 10000 sided polygon is also a convex polygon.

Hopefully, you’re now intrigued enough to learn about more different types of polygons.

What is the Difference Between Rational and Irrational Numbers

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If you’re unsure about the difference between rational and irrational numbers, keep reading to discover a simple guide to understanding the key differences between rational and irrational numbers.

What is the difference between rational and irrational numbers?

The primary difference between irrational numbers and rational numbers is that a rational number can be easily expressed as a whole fraction.

A rational number can always be turned into a fraction, which has both a full numerator number and a full denominator number. While an irrational number can’t be expressed as a whole fraction and can’t be written down as a simple ratio of two integers.

Are there any other noticeable differences between rational and irrational numbers.
Yes, while irrational numbers can’t be written down as simple, whole fractions, irrational numbers can easily be expressed as decimal numbers.
Furthermore an irrational number features endless non repeating digits after it’s decimal point. An example of an irrational number is pi which is most commonly shortened to 3.14159. Keep in mind that it is impossible to express a never ending decimal number such as pi with a simple fraction.
To cement the knowledge that you’ve just read, another example of a recurring infinite decimal number, which can be referred to accurately as a rational number is the golden ratio, which is sometimes shortened to 2.7182818.
Everything you need to know about the major differences between rational and irrational numbers:

1. There are infinite irrational numbers
You may be surprised to hear that there is at least one irrational number between any two rational numbers.

2. Every whole number is a rational number.

If you’re confused about whether the number zero is a rational number, the answer is yes because zero and any other whole number can be divided by the number one, which makes zero a rational number.

3. Decimal numbers tend to be irrational numbers, however there are exceptions

Decimal numbers such as pi also tend to be irrational numbers as they aren’t whole numbers and therefore can’t be divided using a whole numerator and a whole denominator.

However there are exceptions. As an example, recurring decimal numbers such as 0.26262626 are rational numbers.

If in doubt, remember that decimals which are infinite and don’t feature a number sequence are irrational numbers, while decimal numbers which feature recurring numbers such as 0.26262626 are always classed as rational numbers.

4. Negative whole numbers are also classified as being rational numbers

If you were confused by negative whole numbers and whether they are rational numbers or irrational numbers the answer is that negative numbers such as -6 can still be easily divided by themselves and therefore can be expressed by whole fractions and are rational numbers.

5. Square roots can also be irrational numbers

Another example of a common irrational number which has not been discussed above are square roots, which can also be irrational numbers.

After reading the definitions and tips which were listed above, you should now be confident about being able to class both rational numbers and irrational numbers.

How to Save Money: The Easy Way

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In order to discover how to save money without a whole lot of effort, continue reading to discover some of the simplest ways to save money!

How to save money: The easy way

1. Remember to budget for incidental items that change from month to month

Many individuals fail to budget for incidental items that aren’t part of their regular monthly budgets, which can place a strain on their budgets. So ensure to factor in incidental costs into your monthly budget. As an example, make sure to budget money to purchase your friends and family members birthday gifts.

2. Don’t give in to the urge to impulse shop

If you’re often tempted to purchase items on a whim, it’s a great idea to avoid impulse shopping. Next time you find yourself tempted to purchase an item which you saw in a shop window or on one of your favorite online websites, wait at least 24 hours before purchasing the item which you have in mind.

As you’ll normally find that within 24 hours, you’ll think twice and will decide that you don’t really need the item that you were ready to blow your budget, to purchase.

3. Borrow items from friends and family members, instead of purchasing them

As an example, if you’re looking to go on a road trip and are considering purchasing items such as a bicycle, a tent and a blow up mattress, it’s well worth asking your close friends and family members whether they have some of the items which you need. As most of your loved ones will be more than willing to help you out and to loan you the items which you require.

4. Don’t be afraid to purchase pre loved items

One way to save thousands of dollars each year is to purchase pre loved items such as clothes and accessories and furniture. As an example, you can often get great deals on items that you’re looking for on online auction sites such as eBay and at local yard sales and thrift stores.

If you’re worried about your friends judging you for buying preloved items, the truth is that your friends probably won’t be able to tell that you’re purchasing preloved items, especially if you purchase items which are in great condition.

If you’re lucky you may even come across items that have never been used before at a fraction of their original recommended retail price.

5. Choose one day a week, where you don’t spend money

It’s a great idea to choose one night a week where you rely on free activities to entertain yourself rather than spending money at a movie theatre or going out for drinks with friends.

As an example, you may enjoy taking a walk along a nearby beach or playing card games or board games with your friends or family. Once you discover how much fun you can have without spending money, you may find it easier to spend less money the rest of the week.

So if you’ve been on the lookout for easy ways to save money, it’s well worth getting started with the five money saving tips listed above.

How to be Successful: A Fool Proof Guide

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If you’re focused on achieving your goals and are looking for a simple yet effective guide to becoming successful, you’ve come to the right place.

How to be successful: A fool proof guide

1. Write a detailed plan of the actions which you need to take to meet your goal

Successful individuals usually have one thing in common, they plan to succeed. One of the best ways to plan to succeed and to meet your goals is to ensure that you have a detailed plan of action to follow. Included in your plan should be short term goals, which you should achieve each month and each week, in order to get closer to meeting your ultimate long term goal.

2. Talk to individuals who have already obtained the success which you plan to achieve

In order to avoid some of the pitfalls on your route to your goal, it’s well worth arranging to talk to a couple of individuals who have already obtained the success which you plan to achieve yourself.

As an example, if you’re interested in getting into business, it’s well worth tracking down an experienced business mentor, who’ll be able to help you avoid making unnecessary mistakes, while providing you with tips and tricks, which may help you achieve your goals a little faster.

3. Don’t be afraid to ask for help

Contrary to popular belief you don’t have to achieve success entirely on your own merit. While you won’t be successful unless you give 100% of your energy, focus and passion to your goals and endeavours, most successful individuals aren’t afraid to ask from help and to obtain information and advice from to others.

4. Make sure to ask the right individuals for help

Just be sure to ask the right individuals for assistance. As an example, if your goal is to run your first marathon, it may be worth reaching out to a running coach and a dietician, to help you meet your goals but it may not be a wise idea to ask for assistance from negative friends or family members, who may not believe that you can achieve your goal of running a marathon.

5. Identify your strengths and weaknesses

By identifying your strengths, you’ll be able to use your innate strengths to your advantage and if you successfully identify your weaknesses, you’ll be able to work on turning the skills which you’re currently weak at into strengths.

6. Don’t be afraid to alter your plans

As you start taking actions to meet your goals, you may find that you may need to alter your original plans from time to time. So, just make sure that you’re not rigid when it comes to following plans and that you have enough sense to be able to adapt your plans on your journey towards success.

7. Celebrate every success

Instead of waiting until you’ve reached your end goal, increase your motivation to meet your ultimate goal by remembering to mentally celebrate each success and milestone which you tick off!

So no matter what your goal or passion is, it’s well worth using the tried and tested tips listed above, which will help you get on track to meeting your goals.

6 Business Marketing tips that are well worth Trying Out

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In order to discover a few new marketing techniques will can increase your business’ sales and profit, simply continue reading.

6 Business marketing tips that are well worth trying out:

1. Reuse information from your previous marketing campaigns

As an example, you may want to reuse information from one of your business’ blog pieces to create a marketing video. Or you may want to reuse well shot images of your business’ products and services which you’ve posted on your website, to create new social media posts for your business’ various social media accounts on Twitter, Facebook and Instagram.

2. Use competitions to attract attention

An easy way to attract attention to your business’ social media accounts is to use simple competitions, to encourage individuals to follow your accounts or to make purchases on your business’ website.

As an example, you may may want to run a competition which offers individuals the chance to win a new product which is due to be released in the coming weeks or you may want to offer your customers the chance to win the back the value of their last purchase with your business. You may even stipulate that entrants must follow your business’ social media pages or sign up to an email subscription list.

As most individuals have a tendency to be tempted to win something for nothing, you should have no trouble attracting plenty of entries. You may even be interested in offering double entries for individuals who share your original social media post on their own social media account.

3. Try to win an online business award to post on your social media pages and website

If you manage to win an online business award in your industry, you may may be able to advertize your award on your social media accounts and your website. If winning an established business award seems a little far fetched for your business, you may want to start up your own business awards for businesses in your city or area.

As a bonus, if you start your own business awards, you may form valuable industry contacts with other businesses in your industry who aren’t in direct competition with your business and offer different services or products to your business.

4. Carefully use guerrilla marketing techniques

If you haven’t heard of the term guerrilla marketing, it involves using street advertising to market your business’ online presences. As an example, you may want to use chalk to write your business’ social media handles on the pavement outside your real life store. Or you may want to put up give out business cards that have your business’ website and social media links, printed on them.

5. Create a monthly or weekly newsletter

One way to keep your customers well informed of new products and services as well as sales is to create a monthly or weekly newsletter, which you can encourage your customers and followers to sign up to.

6. Make sure that your emails and newsletters offer valuable content

If your emails and newsletters offer valuable content, individuals will be far less likely to unsubscribe from your business’ marketing emails.

Hopefully you’ve discovered at least one new marketing technique which you believe can increase your business’ sales and profits.

7 Business Budgeting Tips

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In order to take control of your business’ finances, it’s a great idea to work on budgeting for your business’ future.

7 Business budgeting tips:

1. Overestimate your business’ operating costs

One of the worst mistakes which you can make, when it comes to creating a realistic, workable budget for your business is to underestimate your business’ operating costs. Remember that it’s always a better idea to slightly overestimate your costs than to underestimate your business’ costs and to run out of funds to run each area of your business effectively.

2. Make sure that your business has an emergency fund available

It’s well worth ensuring that your business has a healthy emergency fund, which is kept seperate from your business’ day to day accounts. Having an emergency fund available, may come in handy if any unexpected costs arise or if you want to upgrade your business and to take your business to the next level.

3. Make sure that you’re not getting overcharged by your current supplier

Even if your business has a long term supplier, who you’re used to dealing with, it’s always a smart idea to get quotes from other suppliers who may be able to provide your business with the same quality product as your current supplier for a much lower price.

4. Try and predict how your sales numbers may change in the next 12 months

One way to create an accurate budget is to try and predict how your sales numbers may change over the next 12 months. As an example, when you look at your business’ sales statistics for the last 12 months you may notice that your sales increase in the months that lead up to Christmas or that you sell more products or services during a particular season such as summer.

5. Save a set percentage of your profit

It’s well worth deciding on a set percentage of your profit, which you’ll save in your business’ emergency fund, each year. If your profits are particularly healthy one year, you may also want to increase the amount of money which you save for a rainy day or to take your business to the next level.

6. Make sure to work on paying off your business loans

Your end goal should be to have zero debt. As you know it’s also wise to pay off your loans as quickly as possible, in order to avoid paying high levels of interest on your business loans. So it’s a great idea to invest a fair amount of your business’ profits in to getting on top of your business debt.

7. Make sure to review your business’ budget on a monthly basis

instead of setting a year long budget and not looking at your budget for 12 months, it’s a wise idea to review your business’ budget on a month to month basis. As you may have to tweak your budget, multiple times over a calendar year.

So if you’re a business owner who is looking to budget for the financial future of your business, it’s a great idea to implement many of the budgeting tips listed above!

Travel and education: Study abroad tips

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If you’re planning a study abroad trip, continue reading to discover a few handy tips that will make your time overseas a whole lot easier!

Travel and education: Study abroad tips

1. Purchase a journal to write about your daily experiences living abroad

It’s well worth going out and purchasing a journal which you’ll be able to pack for your study abroad experience. As while you can write about your daily experiences as an international student using a digital diary, as the years go by, you’ll love being able to read a physical diary of your experiences abroad.

You may want to place trinkets such as ticket stubs to sports game or concerts which you attend as well as polaroid photos of yourself with your new friends in your travel journal.

2. Start a blog to document your experience

In order to keep all of your friends and family members back home updated on your experience, you may want to consider starting a blog about your experience. Just be sure to post plenty of photos to accompany your posts as your loved ones will be excited to see photos of you living abroad and in the years to come, you’ll also enjoy looking back at your blog posts.

3. Set yourself a weekly budget in order to manage your expenses

in order to avoid running out of disposable cash on your study abroad experience, it’s a great idea to set yourself a weekly budget, which you’ll be able to realistically stick to for the duration of your time abroad. As so many international students waste a large portion of the money which they’ve saved for their trip in their first month abroad.

4. Make sure to have digital copies of all of your important documents

It’s well worth scanning or taking clear smart phone photos of your passport, your student exchange documents and your visas as it always pays to be safe, rather than sorry. Especially, when you’re traveling alone to a foreign country, which you’ve never visited before.

5. Use Skype or Face Time to talk to your loved ones back home

If you ever get homesick, simply video chat with your friends and family members back home, to hear familiar voices without having to pay for expensive international phone calls. Just make sure that you know what time it is in your home city, before calling your loved ones to avoid waking them up in the middle of the night or early in the morning.

6. Take a small trinket with you on your adventure, to remind you of home

Another way to fight homesickness is to pack a small item which reminds you of home, during your study abroad experience. As an example, you may want to pack your favorite mug that your best friend brought you for Christmas or your favorite pillow case or blanket.

7. Join a variety of clubs in order to make new friends

As an example, you may want to join a hiking or skiing club that embarks on regular weekends away.

So if you have your heart set on studying abroad in the future, it’s well worth keeping all the tips listed above in mind.