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100 Day Challenge 2020

Exponents, Simplified

When we see repeated addition such as 3+3+3+3+3,3 + 3 + 3 + 3 + 3, we can rewrite the expression and make it shorter with multiplication. In this case, we get 5×3.5 \times 3. Meanwhile, we can make an expression with repeated multiplication more compact with an exponent. For example, we can rewrite 4×4×44 \times 4 \times 4 as 43.4^3.

What happens when we see a combination of repeated operations in one expression, such as 43+43+43+43?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 43+43+43+43.4^3 + 4^3 + 4^3 + 4^3. The term 434^3 appears four times in the repeated addition, so we can think of it as being multiplied by 4.4. So,

43+43+43+43=4×43.4^3 + 4^3 + 4^3 + 4^3 = 4 \times 4^3.

Is there now some way we can combine the 4’s?4\text{'s?} Let's look at the factor 434^3 first. It means multiplication of three factors of 4,4, so we can now rewrite the expression as

4×43=4×(4×4×4)=4×4×4×4.\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 44.4^4. So, 43+43+43+43=44.4^3 + 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 3634?3^6 - 3^4? Can we combine the two terms into one? Let's start by thinking about what 363^6 and 343^4 mean. Each represents repeated multiplication of factors of 3.3. Specifically,

3634=(3×3×3×3×3×3)(3×3×3×3). 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 3.3. Let's factor out four factors of 33 from each term. This means we are factoring 343^4 from each term:

3634=34(36343434)=34(3×3×3×3×3×33×3×3×33×3×3×33×3×3×3). \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 33 cancels multiplication by 3,3, so the divided terms simplify:

34(3×3×3×3×3×33×3×3×33×3×3×33×3×3×3)=34((3×3)1). \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 (3×3)1=91=8,(3 \times 3) - 1 = 9 - 1 = 8, so we can rewrite 34((3×3)1)=8×34. 3^4 \big( (3 \times 3) - 1 \big) = 8 \times 3^4. So,

3634=8×34. 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?

3100+3100+310031013100399 \frac{3^{100} + 3^{100} + 3^{100} }{3^{101} - 3^{100} - 3^{99}}

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