Question

order the expressions from what you think is least complicated to most complicated.
least complicated
$\frac{7^3 cdot 7^3}{7^2}$
$\frac{7}{7} cdot \frac{7}{7} cdot \frac{7}{7} cdot 7 cdot 7 cdot 7 cdot 7$
$\frac{7^7}{7^3}$
$7^4$
$\frac{7 cdot 7 cdot 7 cdot 7 cdot 7 cdot 7 cdot 7}{7 cdot 7 cdot 7}$

Explanation:

Step1: Simplify each expression

• $7^1 = 7$
• $\frac{7^7}{7^3}=7^{7-3}=7^4=2401$
• $\frac{7}{7} \cdot \frac{7}{7} \cdot \frac{7}{7} \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 1 \cdot 1 \cdot 1 \cdot 7^4=7^4=2401$
• $\frac{7^3 \cdot 7^3}{7^2}=7^{3+3-2}=7^4=2401$
• $\frac{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7}=\frac{7^7}{7^3}=7^{7-3}=7^4=2401$

Step2: Rank by complexity

Complexity is based on how directly the expression represents its value, with simpler forms having fewer operations or condensed notation.

1. $7^1$ (single term, no extra operations)
2. $\frac{7^7}{7^3}$ (simple exponent rule application)
3. $\frac{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7}$ (expanded but straightforward division)
4. $\frac{7^3 \cdot 7^3}{7^2}$ (two exponent multiplications first)
5. $\frac{7}{7} \cdot \frac{7}{7} \cdot \frac{7}{7} \cdot 7 \cdot 7 \cdot 7 \cdot 7$ (multiple redundant 1s from division)

Answer:

LEAST COMPLICATED

1. $7^1$
2. $\frac{7^7}{7^3}$
3. $\frac{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7}$
4. $\frac{7^3 \cdot 7^3}{7^2}$
5. $\frac{7}{7} \cdot \frac{7}{7} \cdot \frac{7}{7} \cdot 7 \cdot 7 \cdot 7 \cdot 7$

MOST COMPLICATED