Question

What is the value of ${{64}^{\dfrac{5}{3}}}?$

Answer 1

Hint: We are going to use the concept of exponents to solve this question. We convert the base into a product of prime factors and use the formula of exponents which is ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}.$ This is known as the power law of exponents. Using this, we calculate the value of the term ${{64}^{\dfrac{5}{3}}}.$

Complete step-by-step solution:
To solve this question, let us first represent the number 64 which is the base, in terms of the product of prime factors. It can be represented solely by the power of 2. This is given as
$\Rightarrow 64=2\times 2\times 2\times 2\times 2\times 2$
Any number x multiplied with itself n number of times can be represented in exponents form as ${{x}^{n}}.$
$\Rightarrow 64={{2}^{6}}$
Therefore, we can represent 64 as ${{2}^{6}}.$
Now substituting this in the term given in the question,
$\Rightarrow {{\left( {{2}^{6}} \right)}^{\dfrac{5}{3}}}$
We now use the power law of exponents which is given by,
$\Rightarrow {{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$
Using this for our calculations,
$\Rightarrow {{2}^{6\times \dfrac{5}{3}}}$
Cancelling the power 6 as 2 times 3 with the 3 in the denominator of the exponent,
$\Rightarrow {{2}^{2\times 5}}$
Taking a product of the two exponents,
$\Rightarrow {{2}^{10}}$
Expanding this as the product of 2 ten times,
$\Rightarrow 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
We get the value to be
$\Rightarrow 1024$
Hence, the value of the term ${{64}^{\dfrac{5}{3}}}$ is 1024.

Note: It is important to know the basics of exponents and their laws in order to solve such questions. We also need to know to simplify to the prime factors form. We can also solve this question without converting to prime factors and since we know that 64 can be written as ${{4}^{3}}$ and can be simplified in a similar manner as shown in the solution above.