Question

the value of ⁴√(16)²​

Answer 1

Answer:

ANS)4

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Answer 2

The value of the given expression of root and power is 4.

Here, the 4 before the root sign denotes to finding the n-th root i.e. 4th root. In general 4th root of a number "X" is a number "Y" such that multiplying it 4 times gives the result X i.e.

                                       $\sqrt[4]{x} = y$ ⇒ $y * y*y*y = x$

Here, we first find the square of 16 and then the 4th root of the result.

                                  $\sqrt[4]{16^{2} }$

                             $= \sqrt[4]{16*16 }$

                             $= \sqrt[4]{256 }$

                 

For finding the 4th root of 256, we first do the prime factorization of it and then group factors used 4 times in the answer.

Prime factorization of 256 :

$256 = 2 * 2 * 2 *2*2*2*2*2$          ( i.e. 2 multiplied 8 times )

      $= ( 2*2*2*2) * (2*2*2*2)$.   ( combining in sets of 4 commons )

      $= ( 2^{4} * 2^{4} )$

                $\sqrt[4]{256 } = \sqrt[4]{(2*2*2*2)*(2*2*2*2) }$

                         $= \sqrt[4]{2^{4} * 2^{4} }$

                         $= \sqrt[4]{(2*2)^{4} }$

                         $= \sqrt[4]{4^{4} }$

                         $= 4$    ( as 4th power and 4th root cancel each other )

Hence, the value of the given expression evaluates to 4.

To learn more about Powers and Roots, visit

https://brainly.in/question/29106946

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