Question

Write in exponential form: ${{2}^{3}}+{{2}^{5}}+{{2}^{6}}-{{2}^{2}}$

Answer 1

Hint: Start by taking ${{2}^{2}}$ common from all the terms present in the expression followed by putting the values of powers of 2 present inside the bracket. Once you put the values and solve it again convert it to the simplest form as a power of 5 and use the identity ${{a}^{2}}{{b}^{2}}={{\left( ab \right)}^{2}}$ to get the answer.

Complete step by step answer:
Let us start by taking ${{2}^{2}}$ common from all the terms given in the expression that we have to simplify. On doing so, we get
${{2}^{3}}+{{2}^{5}}+{{2}^{6}}-{{2}^{2}}$
$={{2}^{2}}\left( 2+{{2}^{3}}+{{2}^{4}}-1 \right)$
Now we know that the value of ${{2}^{3}}=8$ and ${{2}^{4}}=16$ . If we put these values in our expression, we get
${{2}^{2}}\left( 2+8+16-1 \right)$
Now we will simplify the part inside the bracket. On doing so, we get
${{2}^{2}}\left( 26-1 \right)$
$={{2}^{2}}\times 25$
Now, we know that the value of the square of 5 is 25. If we use this in our equation, we get
${{2}^{2}}\times {{5}^{2}}$
Also, we know that ${{a}^{2}}{{b}^{2}}={{\left( ab \right)}^{2}}$ .
${{\left( 2\times 5 \right)}^{2}}$
$={{10}^{2}}$
Therefore, we can conclude that the simplest exponential form of ${{2}^{3}}+{{2}^{5}}+{{2}^{6}}-{{2}^{2}}$ is ${{10}^{2}}$ .

Note: In such questions the first thing that should come to your mind is taking common, as it reduces the calculation of higher powers of numbers, because in such questions you mostly need to substitute the values of powers of numbers. Also, try to report the answer in the simplest possible manner as for the above question we could have reported the answer as ${{2}^{2}}\times {{5}^{2}}$ , but this won’t be the simplest form.