Question

Find the value of:
1) ${64^{\dfrac{1}{2}}}$
2) ${32^{\dfrac{1}{5}}}$
3) ${125^{\dfrac{1}{3}}}$

Answer 1

Hint:
A fractional exponent is an alternate way to express the powers and roots together. For example, the following are equivalent.
${81^{\dfrac{1}{2}}} = \sqrt[2]{{81}}$
The numerator denotes the power whereas the denominator denotes the index of the root. If there is no power being applied, write “1” in the numerator as a placeholder, which denotes the index of the root as shown below.
${81^{\dfrac{1}{2}\dfrac{{ \to power}}{{ \to \;\quad root}}}}$

Complete step by step solution:
${64^{\dfrac{1}{2}}}$, the expression states that we need to calculate the square root of 64.
${64^{\dfrac{1}{2}}} = \sqrt[2]{{64}}$
Now we know that $64 = 8 \times 8$
So, which means 64 is a square of 8.
Hence, 8 is a square root of 64.
$ \Rightarrow {64^{\dfrac{1}{2}}} = \sqrt[2]{{64}} = \sqrt[2]{{8 \times 8}} = 8$
Therefore, $ \Rightarrow {64^{\dfrac{1}{2}}} = 8$
${32^{\dfrac{1}{5}}}$, the expression states that we need to calculate the fifth root of 32.
${32^{\dfrac{1}{5}}} = \sqrt[5]{{32}}$
Now we know that $32 = 2 \times 2 \times 2 \times 2 \times 2$
So, 32 is the fifth of 2.
Hence, 2 is a fifth root of 32.
$ \Rightarrow {32^{\dfrac{1}{5}}} = \sqrt[5]{{32}} = \sqrt[5]{{2 \times 2 \times 2 \times 2 \times 2}} = 2$
Therefore, $ \Rightarrow {32^{\dfrac{1}{5}}} = 2$
${125^{\dfrac{1}{3}}}$, the expression states that we need to calculate the cube root of 125.
${125^{\dfrac{1}{3}}} = \sqrt[3]{{125}}$
Now we know that $125 = 5 \times 5 \times 5$
So, which means 125 is a cube of 5.
Hence, 5 is a cube root of 125.
$ \Rightarrow {125^{\dfrac{1}{3}}} = \sqrt[3]{{125}} = \sqrt[3]{{5 \times 5 \times 5}} = 5$

Therefore, $ \Rightarrow {125^{\dfrac{1}{3}}} = 5$

Note:
Please note when it comes to dealing with exponents students often make mistakes. Sometimes, they multiply the base and the exponent. For e.g. ${16^{\dfrac{1}{4}}}$is not equal to 4, it's 2. Then when using the multiplication rule don’t forget it only applies to expressions with the same base. For e.g.: - Four squared times two cubed is not the same as 8 raised to the power two plus three.${4^2} \times {2^3} \ne {8^{2 + 3}}$. The last one is the multiplication rule applies just to the product, not to the sum of two numbers.