## Exponents, Simplified

When we see repeated addition such as $3 + 3 + 3 + 3 + 3,$ we can rewrite the expression and make it shorter with multiplication. In this case, we get $5 \times 3.$ Meanwhile, we can make an expression with repeated multiplication more compact with an exponent. For example, we can rewrite $4 \times 4 \times 4$ as $4^3.$

What happens when we see a combination of repeated operations in one expression, such as $4^3 + 4^3 + 4^3 + 4^3?$ Can we rewrite this expression more simply?

Let's begin by looking at the repeated addition in $4^3 + 4^3 + 4^3 + 4^3.$ The term $4^3$ appears four times in the repeated addition, so we can think of it as being multiplied by $4.$ So,

$4^3 + 4^3 + 4^3 + 4^3 = 4 \times 4^3.$

Is there now some way we can combine the $4\text{'s?}$ Let's look at the factor $4^3$ first. It means multiplication of three factors of $4,$ so we can now rewrite the expression as

\begin{aligned} 4 \times 4^3 & = 4 \times (4 \times 4 \times 4 )\\ & = 4 \times 4 \times 4 \times 4. \end{aligned}

This is simply $4^4.$ So, $4^3 + 4^3 + 4^3 + 4^3 = 4^4.$ By switching between the operations of addition, multiplication, and exponentiation, we can make the expression much simpler.

How about an expression such as $3^6 - 3^4?$ Can we combine the two terms into one? Let's start by thinking about what $3^6$ and $3^4$ mean. Each represents repeated multiplication of factors of $3.$ Specifically,

$3^6 - 3^4 = (3 \times 3 \times 3 \times 3 \times 3 \times 3) - ( 3 \times 3 \times 3 \times 3).$

Each term has at least four factors of $3.$ Let's factor out four factors of $3$ from each term. This means we are factoring $3^4$ from each term:

\begin{aligned} 3^6 - 3^4 & = 3^4 \left( \frac{3^6}{3^4} - \frac{3^4}{3^4} \right) \\ & = 3^4 \left( \frac{3 \times 3 \times 3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3} - \frac{3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3} \right). \\ \end{aligned}

Division by $3$ cancels multiplication by $3,$ so the divided terms simplify:

\begin{aligned} & 3^4 \left( \frac{3 \times 3 \times 3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3} - \frac{3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3} \right) \\ & = 3^4 \big( (3 \times 3) - 1 \big). \\ \end{aligned}

We can calculate $(3 \times 3) - 1 = 9 - 1 = 8,$ so we can rewrite $3^4 \big( (3 \times 3) - 1 \big) = 8 \times 3^4.$ So,

$3^6 - 3^4 = 8 \times 3^4.$

Are there any exponent rule shortcuts we could have used to make our simplifying more efficient?

# Today's Challenge

What does this expression simplify to?

$\frac{3^{100} + 3^{100} + 3^{100} }{3^{101} - 3^{100} - 3^{99}}$ 