Question

Find the value of$x$, if ${2^{5x}} \div {2^x} = \sqrt[5]{{{2^{20}}}}$

Answer 1

In our question we have to know
certain rules on index which we are going to use to solve the problem.
We know that, $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
And,
$\sqrt[p]{{{a^q}}} = {a^{\dfrac{q}{p}}}$
Using the formulas, we will divide the given terms and compare both the sides. After comparison we can find the value of$x$.

Complete step-by-step answer: It is given that ${2^{5x}} \div {2^x} = \sqrt[5]{{{2^{20}}}}$
We have to find the value of $x$.
With the certain rules for indices we are going to solve the given equation
We know that, $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ and, $\sqrt[p]{{{a^q}}} = {a^{\dfrac{q}{p}}}$
Let us consider the given question,
${2^{5x}} \div {2^x} = \sqrt[5]{{{2^{20}}}}$
Using $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$,
${2^{5x - x}} = \sqrt[5]{{{2^{20}}}}$
Using $\sqrt[p]{{{a^q}}} = {a^{\dfrac{q}{p}}}$,
${2^{5x - x}} = {2^{\dfrac{{20}}{5}}}$
Let us simplify the above equation we get,
${2^{5x - x}} = {2^{\dfrac{{20}}{5}}}$
By simplifying the terms in the power by subtracting and dividing, we get,
${2^{4x}} = {2^4}$
We know that, if the bases are equal, then the power has to be equal.
With this condition we compare the above equation,
So, we can come to a conclusion that, $4x = 4$
Let us divide by 4 on both sides and simplifying we get,
$x = 1$
Hence, we have found the value of.
The value of $x$ in the given equation is 1.

Note: Index or indices of a number means how many times to use the number for multiplication.
For example, ${a^m}$ means $a$ is multiplied to itself for $m$ times.
We know that, if the bases are equal, the power has to be equal.
i.e. ${a^n} = {a^m} \Rightarrow m = n$ .
If a number is taken ${n^{th}}$ root, it can be said that the number is multiplied to itself for the reciprocal of n times.
i.e. it can be expressed as follows $\sqrt[n]{a} = {a^{\dfrac{1}{n}}}$