Question

find the value of 4th root of √64^-2​

Answer 1

Answer:

1/√8

Step-by-step explanation:

4th root of √64^-2​ = (√64)^-(2/4)

                               = 64^(-2/(4 x 2))

                               = 64^(-2/8)

                               = 1/64^(1/4)

64 = 2^6

∴ 1/64^(1/4) = 1/2^(6/4)

                  = 1/2^(3/2)

                  = 1/√8

Hope this helps! :)

Answer 2

Given:

($\frac{1}{{64} }$)²

To find:

The fourth root of the given number

Solution:

• The "root of a number x is a number which when multiplied with itself, equals x".
• The fourth root of a number x means finding $\sqrt[4]{x}$.
• For finding the solution, we will follow these steps:

1. Simplify the given number

• The number given to us is:

                               ( $\frac{1}{{64} }$)²

                          ⇒ $\frac{1}{4096}$

                                   

2. Find its 4th root.

                            $\sqrt[4]{\frac{1}{4096} }$

                           ⇒$\frac{1}{4096} ^{1/4}$

• 4096 can be expressed as the square of 64.
• 64 can be expressed as the square of 8.

                    4096 = (64)² = [(8)²]² = (8)⁴

• Thus, we get:

                                       $\frac{1}{8^{4} } ^{1/4}$

• The powers get multiplied and cancel each other out.

                                   ⇒  $\frac{1}{8}$                                    (1ⁿ = 1)

Therefore, the 4th root of ($\frac{1}{64}$)² is $\frac{1}{8}$.