Question

What is the value of y in the product of powers below?

Answer 1

The value of y = 0

Explanation:

Given the equation

`8^3\cdot 8^(-5)\cdot 8^(\:y)=8^(-2)`

Apply exponent rule:
`a^b\cdot \:a^c=a^(b+c)`

so

`8^3\cdot \:\:8^(-5)=8^(3-5)=8^(-2)=(1)/(8^2)=(1)/(64)`

Thus, the equation becomes

`(1)/(64)\left(8\:^y\right)=(1)/(64)`

Divide both sides by 1/64

`((1)/(64)\left(8\:^y\right))/((1)/(64))=((1)/(64))/((1)/(64))`

`8\:^y=1`

Solve Exponent

`8\:^y=1`

Take log of both sides

`\:log\left(8\:^y\right)\:=log\:\left(1\right)`

`y\cdot \left(log\left(8\right)\right)=log\left(1\right)`

`y\:=(log\left(1\right))/(\left(log\left(8\right)\right))`

`y = 0`

Therefore, the value of y = 0