Question

Find the missing power in the following calculation:

$5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot 5^x=5$

Short answer

The missing power in the expression$5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot 5^x=5$ is $\frac{2}{3}$.

Step-by-step solution

Step 1. Laws of Exponents.

• $a^m\cdot a^n=a^{m+n}$

• $a^m=a^n\iff m=n$
  

Step 2. Evaluate the given expression.

Given$5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot 5^x=5$

We need to find x.

Step 3. Find the missing value.

Consider$5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot 5^x=5$

LHS is$5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot 5^x=5^{\frac{1}{3}+x}$

RHS is$5^1$

Equate LHS and RHS

$5^{\frac{1}{3}+x}=5^1$

Since the base is the same, we can equate the exponents,

$\begin{array}{l}\frac{1}{3}+x=1\\ \Rightarrow x=1-\frac{1}{3}\\ \Rightarrow x=\frac{2}{3}\end{array}$

So, the missing power is$\frac{2}{3}$