Question

What is $8$ to the ${{5}^{th}}$ power?

Answer 1

Hint: We know that the power of a number is another word for the exponent of a number. If a number is raised to another number, then that means we multiply the former the latter times. We can interpret this Mathematically as ${{a}^{b}}=a\times a\times ...\times {{a}_{\left( b times \right)}}.$

Complete step by step solution:
Let us consider the given statement, $8$ to the ${{5}^{th}}$ power.
By the phrase ‘to the ${{5}^{th}}$ power’, we mean we need to multiply the base with itself $5$ times.
We can write this word phrase Mathematically as ${{8}^{5}}.$
So, our base is $8$ and we should find the product of five $8.$
If we are interpreting this Mathematically, we will get the following identity, ${{a}^{b}}=a\times a\times ...\times {{a}_{\left( b times \right)}}.$
Here, in our case $a=8$ and $b=5.$
Now, let us find the product of five $8.$
By the above identity, we will get ${{8}^{5}}=8\times 8\times 8\times 8\times 8.$
We know that ${{8}^{2}}=8\times 8=64.$
From this we will get ${{8}^{5}}=64\times 64\times 5.$
When we consider the right-hand side of the above equation, we can see that there are two $64.$ We can write it using the exponent as ${{64}^{2}}.$
Then we will get ${{8}^{5}}={{64}^{2}}\times 5.$
Since ${{64}^{2}}=4096.$
Now, we will get ${{8}^{5}}=4096\times 5.$
And from this, we will get ${{8}^{5}}=20480.$
Hence the given statement $'8$ to the ${{5}^{th}}$ power’ can be written as ${{8}^{5}}=20480.$

Note: Since we have the identity ${{a}^{b}}=a\times a\times ...\times {{a}_{\left( b times \right)}},$ we can derive another identity from this identity. $\dfrac{{{a}^{b}}}{a}=\dfrac{a\times a\times ...\times {{a}_{\left( b-1 times \right)}}\times {{a}_{ \left( b times \right)}}}{a}=a\times a\times ...\times {{a}_{\left( b-1 times \right)}}.$ From this we will get $\dfrac{{{a}^{b}}}{a}={{a}^{b-1}}.$ This can be generalized to form the following identity $\dfrac{{{a}^{b}}}{{{a}^{r}}}={{a}^{b-r}}.$ As multiplication is repeated addition, division is repeated subtraction. We have obtained $8$ to the ${{5}^{th}}$ power. Similarly, we can find the ${{5}^{th}}$ root of a number.