Question

How do you rewrite the exponential expression as a radical expression ${b^{ - \dfrac{4}{7}}}$?

Answer 1

In this question we have an exponential expression which has a negative power which is in terms of a fraction therefore, we will use two exponential properties to simplify the expression and write it in the radical form.

Formula used:
$\dfrac{1}{{{a^b}}} = {a^{ - b}}$
${a^{\dfrac{b}{c}}} = {({a^b})^{\dfrac{1}{c}}} = \sqrt[c]{{{p^b}}}$

Complete step-by-step answer:
We have the expression as:
$\Rightarrow {b^{ - \dfrac{4}{7}}}$
Now since the exponent is in the negative we know, the inverse of the number, also called the reciprocal of the number is the number dividing $1$ , for example the reciprocal of is $\dfrac{1}{a}$ , and it can also be expressed in terms of power as ${a^{ - 1}}$.
Therefore, on using this property, we get:
$\Rightarrow \dfrac{1}{{{b^{\dfrac{4}{7}}}}}$
Now the denominator is in term of exponent of a fraction, on using the formula ${a^{\dfrac{b}{c}}} = {({a^b})^{\dfrac{1}{c}}} = \sqrt[c]{{{p^b}}}$, we get:
$\Rightarrow \dfrac{1}{{{{({p^4})}^{\dfrac{1}{7}}}}}$
Which could be written as:

$\Rightarrow \dfrac{1}{{\sqrt[7]{{{b^4}}}}}$, which is the required solution.

Note:
It is to be noted that the function can be read out as ‘$1$ divided by the $7th$ root of $b$ raised to $4$’. This means that whatever is the solution when the term $b$ is raised to the power of $4$, we have to take the seventh root of the number, which means the number which has to be multiplied $7$ times to get ${b^4}$.
Exponents are used to write the similar terms in multiplication in a simple format. Some exponential numbers are too big to be calculated directly, therefore to solve them logarithm is used.
Logarithm is used in exponents to convert the exponential term in the form of multiplication. The antilog of the term is then taken which is the inverse of taking log, to get the required solution back after simplification.