Square root: Square root of a number is the value, when multiplied by itself, gives the number under the square root. Thus, [latex]\sqrt{25}[/latex] = 5
Example 1 Evaluate [latex]\sqrt{6084}[/latex] by factorization method.
Solution:

Method: Express the given number as the product of prime factors. Now, take the product of these prime factors choosing one out of every pair of the same primes. This product gives the sqare root of the given number.
Thus; resolving 6084 into prime factors, we get:
6084 = [latex]2^{2}[/latex] x [latex]3^{2}[/latex] x [latex]13^{2}[/latex]
Therefore; [latex]\sqrt{6084}[/latex] = (2 x 3 x 13) = 78.

Example 2 The square root of ([latex]272^{2}[/latex] - [latex]128^{2}[/latex]) is:
Solution:

[latex]\sqrt{(272)^{2} - (128)^{2}}[/latex] = [latex]\sqrt{(272 + 128)(272 - 128)}[/latex] = [latex]\sqrt{400 \times 144}[/latex] = [latex]\sqrt{57600}[/latex] = 240.

Example 3 The square root of [latex]0.\overline{4}[/latex] is:
Solution:

[latex]\sqrt{0.\overline{4}}[/latex] = [latex]\sqrt{\frac{4}{9}}[/latex] = [latex]\frac{2}{3}[/latex] = 0.666..... = [latex]0.\overline{6}[/latex]

Cube root: Cube root of a number is the value, when used in multiplication three times, gives the number under the cube root. Thus, [latex]\sqrt[3]{64}[/latex] = 4
Example 1 Find the cube root of 2744.
Solution:

Method: Resolve the given number as the product of prime factors and take the product of prime factors, choosing one out of three of the same prime factors. Resolving 2744 as the product of prime factors, we get:
2744 = [latex]2^{3}[/latex] x [latex]7^{3}[/latex].
Therefore, [latex]\sqrt[3]{2744}[/latex] = 2 x 7 = 14.

Example 2 By what least number 4320 be multiplied to obtain a number which is a perfect cude?
Solution:

Clearly, 4320 = [latex]2^{3}[/latex] x [latex]3^{3}[/latex] x [latex]2^{2}[/latex] x 5.
To make it a perfect cube, it must be multiplied by 2 x [latex]5^{2}[/latex] i.e, 50.

Example 3 Find the cube root of [latex]\sqrt[3]{4\frac{12}{125}}[/latex].
Solution:

[latex]\sqrt[3]{4\frac{12}{125}}[/latex] = [latex]\sqrt[3]{\frac{512}{125}}[/latex] = ([latex]\frac{8 \times 8 \times 8}{5 \times 5 \times 5})^{1/3}[/latex] = [latex]\frac{8}{5}[/latex] = 1[latex]\frac{3}{5}[/latex].

1. [latex]\sqrt{ab}[/latex] = [latex]\sqrt{a}[/latex] x [latex]\sqrt{b}[/latex]
2. [latex]\sqrt{\frac{a}{b}}[/latex] = [latex]\frac{\sqrt{a}}{\sqrt{b}}[/latex] ⇒[latex]\sqrt{\frac{a}{b}}[/latex] = [latex]\frac{\sqrt{a}}{\sqrt{b}}[/latex] x [latex]\frac{\sqrt{b}}{\sqrt{b}}[/latex] ⇒[latex]\sqrt{\frac{a}{b}}[/latex] = [latex]\frac{\sqrt{ab}}{\sqrt{b}}[/latex]
3. The value of [latex]\sqrt{X} \mp \sqrt{X} \mp \sqrt{X} \mp \dotsb \infty[/latex] Suppose ‘a’ and ‘b’ are consecutive factors of x where b > a. If there sign is (+) in the expression, the answer is b (i.e) bigger factor and if there sign is (-), the answer is a i.e., the smaller factor.
Example: [latex]\sqrt{20} - \sqrt{20} - \sqrt{20} - \sqrt{20} - \dotsb \infty[/latex] is equal to.
Solution: 20 = 4 × 5 Since, the sign is (-), then the required answer is the smaller factor i.e., 4
4. [latex]\sqrt{X} . \sqrt{X} . \sqrt{X} . \sqrt{X} . \dotsb . \infty[/latex] If the root goes upto ∞ in multiplication, the answer is x itself.

1. Evaluate [latex]\sqrt{13641}[/latex] ?
Solution:

Consider [latex]\sqrt{13641}[/latex]
As 13641 = 121 x 121
Therefore [latex]\sqrt{13641}[/latex] = 121

2. What is the square root of 0.00064 ?
Solution:

Consider square root of 0.00064
i.e.[latex]\sqrt{0.00064}[/latex]
⇒[latex]\sqrt{\frac{64}{10000}}[/latex]
⇒[latex]\frac{\sqrt{64}}{\sqrt{10000}}[/latex]
⇒[latex]\frac{8}{100}[/latex]
⇒0.08
Therefore, [latex]\sqrt{0.00064}[/latex] = 0.08

3. If [latex]\sqrt{{1} + {\frac{x}{121}}}[/latex] = [latex]\frac{12}{11}[/latex]then find the value of [latex] x [/latex]?
Solution:

Given that
[latex]\sqrt{{1} + {\frac{x}{121}}}[/latex] = [latex]\frac{12}{11}[/latex]
Now take square root to other side
⇒[latex]({1} + {\frac{x}{121}})[/latex] = [latex]\left(\frac{12}{11}\right)^2[/latex]
⇒[latex]({1} + {\frac{x}{121}})[/latex] = [latex]\frac{144}{121}[/latex]
⇒[latex]\frac{x}{121}[/latex] = [latex]\frac{144}{121}[/latex] - 1
Now take L.C.M. on right hand side i.e. 121
⇒[latex]\frac{x}{121}[/latex] = [latex]\frac{144 - 121}{121}[/latex]
⇒[latex]\frac{x}{121}[/latex] = [latex]\frac{20}{121}[/latex]
By cross multiplication,
⇒[latex] x [/latex] = 20
Therefore, the value of [latex] x [/latex] = 20

4. If [latex]\sqrt{15}[/latex] = 3.88, find the value of [latex]\sqrt{\frac{5}{3}}[/latex]?
Solution:

Given [latex]\sqrt{15}[/latex] = 3.88
Now consider [latex]\sqrt{\frac{5}{3}}[/latex]
multiply with 3 on numerator and denominator
⇒[latex]\sqrt{\frac{5}{3}}[/latex] x [latex]\sqrt{\frac{3}{3}}[/latex]
⇒[latex]\frac{\sqrt{15}}{3}[/latex]
⇒[latex]\frac{3.88}{3}[/latex]
⇒1.29[latex]\bar{3}[/latex]
Therefore [latex]\sqrt{\frac{5}{3}}[/latex] = 1.29[latex]\bar{3}[/latex]

5. Find the cube root of 2744 ?
Solution:

Given
[latex]\sqrt[3]{2744}[/latex]
As 2744 = [latex]2^3[/latex] × [latex]7^3[/latex]
⇒2 x 7
⇒ 14
Therefore, [latex]\sqrt[3]{2744}[/latex] = 14