Question

The value of $\left( {{8}^{-25}}-{{8}^{-26}} \right)$ is
A. $7\times {{8}^{-25}}$
B. $7\times {{8}^{-26}}$
C. $8\times {{8}^{-26}}$
D. None of the above

Answer 1

Hint: Here we have been given an operation between two indices and we have to find its value. Firstly by using the rule of indices ${{a}^{-p}}=\dfrac{1}{{{a}^{p}}}$ we will rewrite our values. Then we will take the LCM and try to cancel out the terms if possible. Finally we will again use the same law and write the denominator as the numerator and get our desired answer.

Complete step by step answer:
We have been given the value as follows:
$\left( {{8}^{-25}}-{{8}^{-26}} \right)$……$\left( 1 \right)$
Now, using the law of indices, if the index is a negative value then we can write it as the reciprocal of the positive index raised to the same variable.
${{a}^{-p}}=\dfrac{1}{{{a}^{p}}}$
Using the above rule in equation (1) we get,
$\left( {{8}^{-25}}-{{8}^{-26}} \right)=\left( \dfrac{1}{{{8}^{25}}}-\dfrac{1}{{{8}^{26}}} \right)$
Next take the LCM as follows,
$\Rightarrow \left( {{8}^{-25}}-{{8}^{-26}} \right)=\left( \dfrac{{{8}^{26}}-{{8}^{25}}}{{{8}^{25}}\times {{8}^{26}}} \right)$

Take ${{8}^{25}}$ common in the numerator we get,
$\Rightarrow \left( {{8}^{-25}}-{{8}^{-26}} \right)={{8}^{25}}\left( \dfrac{8-1}{{{8}^{25}}\times {{8}^{26}}} \right)$
$\Rightarrow \left( {{8}^{-25}}-{{8}^{-26}} \right)={{8}^{25}}\left( \dfrac{7}{{{8}^{25}}\times {{8}^{26}}} \right)$
Cancel the common term from numerator and denominator we get,
$\Rightarrow \left( {{8}^{-25}}-{{8}^{-26}} \right)=\left( \dfrac{7}{{{8}^{26}}} \right)$
Again using the rule ${{a}^{-p}}=\dfrac{1}{{{a}^{p}}}$ we can rewrite the above value as,
$\therefore \left( {{8}^{-25}}-{{8}^{-26}} \right)=7\times {{8}^{-26}}$

Hence the correct option B.

Note: Index is the value by which a variable or constant is raised to a power. It shows the number of times any given number is multiplied. There are some rules or laws that are necessary to understand while solving any problem related to indices. We can’t take the term without using the rule as their power is in negative sign and as we know in negative integers higher number is smaller. For avoiding any mistake while choosing the common term when the power is in negative we change the power in positive and solve the value.