Math Examples of Radicals

Mathematical expressions can be present in any possible form. When Square roots are found in question, we usually solve such problems by removing square roots. On removing square roots we either write ± symbol or we just simply square both sides of equation to remove the roots. To solve a problem that has square root of bkt 7a minus 6 bkt equal to square root of bkt 5a plus 9 bkt solve step by step:
√ (7a - 6) = √ (5a + 9),
Squaring both sides of equation and removing the square root we get:
(√ (7a - 6)) ² = (√ (5a + 9)) ²,
Or ((7a - 6)) ^{2 / 2} = ((5a + 9)) ^{2 / 2,}
Or ((7a - 6)) 1 = ((5a + 9)) 1,
Or (7a - 6) = (5a + 9)......... equation 1.

Here, we are left with a linear equation which can be solved using following method:
MX + c = bX + d,
Where, 'x' is the variable whose value has to be found and M, c, b and d are constants in the equations. Taking the unknowns on one side and known quantities to other side of equation we get:
X (M – b) = d – c,
Or X = (b – c) / (M – b),

Following the same procedure to solve the equation 1, we get:
(7a - 6) = (5a + 9),
Or a (7 – 5) = 9 – (- 6),
Or a * 2 = 9 + 6,
Or a * 2 = 15,
Or a = 15 / 2,

We can substitute the value of 'a' in equation 1 to verify it as follows:
(7 * 15/2 - 6) = (5 * 15/2 + 9),
105/2 – 6 = 75/2 + 9,
93/2 = 93/2,
LHS = RHS.

The Square root of a monomial a ^m x ⁿ, represented as √ a ^m x ⁿ, is the monomial ‘b ^{m / 2} z ^{n / 2}’, which on multiplying with itself gives back the monomial a ^m x ⁿ.
We can state the formerly mentioned statement in mathematical terms as:
It is known that a ^m x ⁿ = b ^{m / 2} z ^{n / 2} * b ^{m / 2} z ^{n / 2}, such that b = √ a and z = √x, we can write that,
√ a ^m x ⁿ = √ (b ^{m / 2} z ^{n / 2} * b ^{m / 2} z ^{n / 2}) = b ^{m / 2} z ^{n / 2}.
Now, we have to compute √ (16y ²⁴).Observe that the number 16 = 4 * 4 and y ²⁴ = y ¹² * y ¹². Thus,
√ (16y ²⁴) = √ (4 * 4 * y ¹² * y ¹²) = 4 * y ¹².
Consequently, we can say that the square root of the monomial 16y ²⁴ is 4 * y ¹².

As we know, the Square root of a number is the reverse of the square of a number. In the same way, to find square root of a monomial is the process to find a monomial which on multiplying with itself gives the other monomial. The sign of square root is ‘√’. Thus, we can symbolize the square root of a monomial term xⁿ as:
√ (x ⁿ) = y ^m.
Such that, y ^m * y ^m = y ^2m = x ⁿ.
We know that the monomial √ k ⁴ is equivalent to √ (k ² * k²).
Hence, √ (k ⁴) = √ (k ² * k²) = k ².
As a result, the square root of k ⁴ is k ².

The Cube root of any number is the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formulae:
³√a³   = ³√ a × a × a = a,
Now first we find the number or value which can be thrice multiply by the same number or expression (or itself). In the above example the number is 35.
³√42875   = ³√ 35×35×35 = 35
The cube root of the 42875 is 35.

The Cube root of any number is the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formulae,
³√a³   = ³√ a × a × a = a,
Now first we find the number or value which can be thrice multiply by the same number or expression (or itself). In the above example the number is 40.
³√64000 = ³√ 40×40×40 = 40;
The cube root of the 64000 is 40.

The Cube roots of any number are the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formulae
³√a³   = ³√ a × a × a = a
Now first we find the number or value which can be thrice multiply by the same number or expression (or itself). In the above example the number is 3p.
³√27p³   = ³√ 3p × 3p × 3p = 3p;
The cube root of the 27 p³ is 3p.   

The Cube roots of any number are the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formula
³√a³   = ³√ a × a × a = a;
Now first we find the number or value which can be thrice multiply by the same number or expression (or itself).
³√17576   = ³√ 26×26×26 = 26;
The cube root of the 17576 is 26.

The Cube root of any number is the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formula;
³√a³   = ³√ a × a × a = a;
Now First we find the number or value which can be the thrice multiply by the same number or expression (or itself). In the above example the number is 7.
³√343   = ³√ 7×7×7 = 7;
The cube root of the 343 is 7.

The Cube root of any number is the value which can be thrice multiply by the same number or expression (or itself). To find the cube root of any number we use the formulae:
³√a³   = ³√ a × a × a = a,
Now first we find the number or value which can be thrice multiply by the same number or expression (or itself). In the above example the number is 2xy².
³√8x³y⁶    = ³√ 2xy²×2xy×2xy² = 2xy²,
The cube root of the 8x³y⁶ is 2xy².  

The Solution of above example is as follows:-
Here we can easily see that we have a radical expression is 7√3 + 5 + 4√3 + 8 which we have to solve.
To solve this expression we start from left to right according to operation rules. We can clearly see that the term 7√3 and 4√3 have same signs and also containing same Square root. So we add both terms
                                                7√3 + 5 + 4√3 + 8,
                                                11√3 + 5 + 8,
Now we again start from left to right and see that we again have two terms 5 and 8 which are same in nature so
                                                11√3 + 13,
So the resulting expression is 11√3 + 13.

The solution of above example is as follows:-
Here we can clearly see that we have a radical expression (1 + √3) (1 - √3) which we have to solve.
As we know from the Algebra that (a + b) (a – b) = a² – b². So
                                                (1 + √3) (1 - √3) = (1)² – (√3)² = 1 – 3 = -2,
Step 3: So the resulting term of expression (1 + √3) (1-√3) is -2.

The Solution of above example is as follows:
 Here we can easily see that we have a radical expression 6√7 - 4√8 - 5√8 + 7√7 which we have to solve.
To solve this expression we start from left to right according to operation rules. We can clearly see that the term 6√7 and 7√7 have same signs and also containing same Square root. So we add both terms
                                                6√7 - 4√8 - 5√8 + 7√7,
                                                13√7 - 4√8 - 5√8,
Now we again start from left to right and see that we again have two terms -4√8 and -5√8 which are same in nature and also contains same square root so we subtract them
                                                13√7 - 9√8,
So the resulting expression is 13√7 - 9√8.

The Solution of above example is as follows:
Here we can easily see that we have a radical expression 3√5 + 2√3 -4√3 + 4√5 which we have to solve.
 To solve this expression we start moving from left to right according to operation rules. We can clearly see that the term 3√5 and 4√5 have same signs and also containing same Square root. So we add both terms
                                                3√5 + 2√3 - 4√3 + 4√5,
                                                7√5 + 2√3 - 4√3,
Now we again start from left to right and see that we again have two terms 2√3 and -4√3 which are not same in nature but contains same square root so we subtract them
                                                7√5 - 2√3,
So the resulting expression is 7√5 - 2√3.

To obtain a number such that on squaring it we get the number 3 515 625, we will have to find all the factors of number 3 515 625. The process is shown below:
5 3515625 5 703125 5 140625 5 28125 5 5625 5 1125 5 225 5 45 3 9   3  
After calculating the factors, we will have to arrange them to obtain the Square root in the following manner:
3 515 625 = 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 3 * 3,
We can rewrite it as 3 515 625 = (5 * 5 * 5 * 5 * 3) * (5 * 5 * 5 * 5 * 3),
Or, 3 515 625 = (1 875) * (1 875),
Thus, we can easily note down the square root as:
√ 3 515 625 = √ (1 875 * 1 875) = 1 875.
Hence, the square root of the number 3 515 625 is the number 1 875.

To obtain the Square root of the number 627 264, we will have to acquire a numeral such that if we square it, we get 627 264. To find it, we will first obtain all the factors of the number 627 264. The factors of 627 264 are calculated as follows:
2 627264 2 313632 2 156816 2 78408 2 39204 2 19602 3 9801 3 3267 3 1089 3 363 11 121   11  
We can write that 627 264 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 * 11 *11,
We can reframe it as 627 264 = (2 * 2 * 2 * 3 * 3 * 11) * (2 * 2 * 2 * 3 * 3 * 11),
Or, 627 264 = (792) * (792),
Thus, we can easily note the square root as:
√ 627 264 = √ (792 * 792) = 792.
Hence, the square root of the number 627 264 is the number 792.