Question

How do you write $ {{64}^{\dfrac{1}{3}}} $ in radical form?

Answer 1

Hint: We will see the definition of exponents. We will see the case where an exponent is a fraction. We will look at the definition of a radical and an example of a radical. We will use this definition to convert the given number with its exponent into its radical form. We will be able to simplify the radical obtained by factorizing it.

Complete step by step answer:
The exponent is defined as the power to which a given number or expression is to be raised. So, in the expression $ {{a}^{b}} $ , the exponent is $ b $ and the base is $ a $ . Now, if the exponent is a fraction, then we get a root of the base. This root corresponds to the denominator of the fraction. So, if we have $ {{a}^{\dfrac{1}{b}}} $ , then we are looking for the $ {{b}^{th}} $ root of $ a $ . A radical is defined to be the symbol that represents this number with a fraction as an exponent. The symbol is $ \sqrt[b]{a} $ which denotes the $ {{b}^{th}} $ root of $ a $ . That means we have $ {{a}^{\dfrac{1}{b}}}=\sqrt[b]{a} $ . As an example, we can look at the cube root of 27. We know that the cube root of 27 is 3. Its radical form is $ \sqrt[3]{27} $ and its exponent form is $ {{27}^{\dfrac{1}{3}}} $ .
The given exponent form is $ {{64}^{\dfrac{1}{3}}} $ . Using the definition, we can write its radical form as $ \sqrt[3]{64} $ . We can simplify this radical form by finding the cube root of 64. We can factorize 64 as
 $ 64=4\times 4\times 4 $
Therefore, the cube root of 64 is 4. That means $ {{64}^{\dfrac{1}{4}}}=\sqrt[3]{64}=4 $ .

Note:
We should be familiar with the meaning of exponents and different rules of exponents. These rules are very useful in simplification while doing calculations. It is important to know how to calculate the square root and cube root of any given number. A factorization is an essential tool for simplifying calculations and also for finding the square roots or cube roots of numbers.