Question

How do you simplify ${16^{ - \dfrac{2}{3}}}$ ?

Answer 1

Hint: In this question, we have been asked how we would simplify the given term. In order to simplify this, we must be aware of exponential rules. First, reciprocate the given number because the power is negative. Then, find the square of the given number. Next step involves finding the cube root of the square that you have just found. In case the perfect cube root doesn’t exist, find the prime factors of the number and then, group the three factors together and write them outside the cube root and write the remaining factors inside the cube root.

Formula used: ${x^{ - n}} = \dfrac{1}{{{x^n}}}$

Complete step-by-step solution:
We are given a number ${16^{ - \dfrac{2}{3}}}$ and we have to simplify the number.
Step 1: In this step, we will reciprocate the number as we will use the rule ${x^{ - n}} = \dfrac{1}{{{x^n}}}$.
$ \Rightarrow {16^{ - \dfrac{2}{3}}} = \dfrac{1}{{{{16}^{\dfrac{2}{3}}}}}$
Step 2: In this step, we will find the square of the number as the power of the number says so.
$ \Rightarrow \dfrac{1}{{{{16}^{\dfrac{2}{3}}}}} = \dfrac{1}{{{{256}^{\dfrac{1}{3}}}}}$
Step 4: Now, we have to find the cube root of $256$.
But, the cube root of $256$ does not exist as a real number
So, we will find the prime factors of $256$.
$ \Rightarrow \dfrac{1}{{{{256}^{\dfrac{1}{3}}}}} = \dfrac{1}{{\sqrt[3]{{256}}}}$
We can write $256 = 4 \times 4 \times 4 \times 4$.
Grouping the factors,
$ \Rightarrow 256 = \left( {4 \times 4 \times 4} \right) \times 4$
Hence, we can write it as –
$ \Rightarrow \dfrac{1}{{\sqrt[3]{{256}}}} = \dfrac{1}{{4\sqrt[3]{4}}}$

Therefore, ${16^{ - \dfrac{2}{3}}} = \dfrac{1}{{4\sqrt[3]{4}}}$.

Note: There are many exponential rules that we must be aware about in order to solve such questions:
Product rule - ${a^x} \times {a^y} = {a^{x + y}}$
Quotient rule - ${a^x} \div {a^y} = {a^{x - y}}$
Power rule - ${\left( {{a^x}} \right)^y} = {a^{xy}}$
Power of a product rule - ${\left( {ab} \right)^x} = {a^x}{b^x}$
This same rule is applicable on division as well
Zero exponent - ${a^0} = 1$
Fractional exponent = ${a^{\dfrac{x}{y}}} = \sqrt[y]{{{a^x}}}$