Question

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

Answer 1

Hint: 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 $\sqrt[n]{a} = {a^{\dfrac{1}{n}}}$