Question

Write ${\left( {16{a^6}} \right)^{\dfrac{3}{4}}}$ in radical form?

Answer 1

Hint: Given the numbers in the form of exponents. First, we will identify the expression for the base and the exponent. Then use the base law of the exponents, multiply the exponent written on the base by the other exponent and reduce the exponent in simplified form. Then write the expression in radical form using the simplified exponent.

Formula used:
The base law of the exponent is given as:
${\left( {{a^x}} \right)^n} = {a^{x \cdot n}}$

Complete step by step solution:
Now, write the radical form of the expression, by writing the denominator of the exponent as root of the radical and the numerator of the exponent as power of the expression inside the radical.
$ \Rightarrow \sqrt[4]{{{{\left( {16{a^6}} \right)}^3}}}$

To simplify the expression, we will apply the formula, $\sqrt {xy} = \sqrt x \cdot \sqrt y $to the expression.
$ \Rightarrow {\left( {16} \right)^{\dfrac{3}{4}}} \times {\left( {{a^6}} \right)^{\dfrac{3}{4}}}$

Write $16$ as a power of $2$.
$ \Rightarrow {\left( {{2^4}} \right)^{\dfrac{3}{4}}} \times {\left( {{a^6}} \right)^{\dfrac{3}{4}}}$
Now, we will multiply the exponent on the base with the other exponent using the base law of exponents.
$ \Rightarrow {\left( 2 \right)^{4 \times \dfrac{3}{4}}} \times {\left( a \right)^{6 \times \dfrac{3}{4}}}$
$ \Rightarrow {\left( 2 \right)^3} \times {\left( a \right)^{3 \times \dfrac{3}{2}}}$

On simplifying the expression, we get:
$ \Rightarrow {\left( 2 \right)^3} \times {\left( a \right)^{\dfrac{9}{2}}}$
Now, write the radical form of the expression, by writing the denominator of the exponent as root of the radical and the numerator of the exponent as power of the expression inside the radical.
$ \Rightarrow {\left( 2 \right)^3} \times \sqrt {{a^9}} $
Now we will find the factors of the radical expression.
$ \Rightarrow {a^9} = {a^2} \times {a^2} \times {a^2} \times {a^2} \times a$

Now, add the radical to the factors of the expression.
$ \Rightarrow \sqrt {{a^2} \times {a^2} \times {a^2} \times {a^2} \times a} $
Now, take out the perfect square factor of the expression.
$ \Rightarrow {2^3} \times {a^2} \times {a^2}\sqrt a $
Multiplying the terms in the expression, we get:
$ \Rightarrow 8{a^4}\sqrt a $

Hence the expression in radical form is $\sqrt[4]{{{{\left( {16{a^6}} \right)}^3}}}$

Note: When a certain number is given, then to simplify the expression, factorize the number using the prime factorization method. Then, find the perfect squares of the number and take out of the radical.