Question

What are the real fourth roots of $ 256? $

Answer 1

Hint: Here we will first of all find the prime factors for the term given and then will make a pair of four same integers in the form of the product to get the fourth root of the given term. Factors are the terms which when multiplied together gives the original number.

Complete step-by-step answer:
Fourth roots of $ 256 $ is expressed as $ \sqrt[4]{{256}} $
Find the prime factors for the given term. Prime numbers are the numbers which has only two factors, that is number one and the number itself. For the prime factors start dividing first by $ 2,3,5,7 $ and so on.
 $ \sqrt[4]{{256}} = \sqrt[4]{{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}} $
To get the fourth power of the terms, make the pair of four terms with the same integers.
 $ \sqrt[4]{{256}} = \sqrt[4]{{\underline {2 \times 2 \times 2 \times 2} \times \underline {2 \times 2 \times 2 \times 2} }} $
The above expression can be written as the power of four
\[\sqrt[4]{{256}} = \sqrt[4]{{{2^4} \times {2^4}}}\]
When powers are the same, bases are written in the product taking the whole power.
\[\sqrt[4]{{256}} = \sqrt[4]{{{{(2 \times 2)}^4}}}\]
Root four can be re-written as –
\[\sqrt[4]{{256}} = {(2 \times 2)^{4 \times \dfrac{1}{4}}}\]
Common factors from the numerator and the denominator cancel each other and therefore remove from the numerator and the denominator.
\[\sqrt[4]{{256}} = (2 \times 2)\]
Simplify finding the product of the terms in the above expression –
\[\sqrt[4]{{256}} = 4\]
This is the required solution.

Note: Remember that the power and exponents are used to write any mathematical expression having repeated terms multiplied in the short form such $ 2 \times 2 \times 2 = {2^3} $ Also remember when the terms are in multiplication then powers are added with the same base while when the terms are in division with the same base then the powers are subtracted.