Simplify Base-7 Expression: 7^(2x+1) × 7^(-1) × 7^x

The Product Rule for exponents states that $a^m \cdot a^n = a^{m+n}$. Think of it this way: $7^2 \cdot 7^3$ means (7×7) × (7×7×7) = 7×7×7×7×7 = $7^5$, which is 2+3!

A negative exponent doesn't change the addition rule! Just treat -1 as a regular number when adding: (2x+1) + (-1) + x = 2x + 1 - 1 + x = 3x.

The parentheses show that the entire expression 2x+1 is the exponent. When adding exponents, treat (2x+1) as one unit: (2x+1) + (-1) + x.

You could rearrange the terms first, like $7^{2x+1} \cdot 7^x \cdot 7^{-1}$, but you'll still get the same answer. The key is always adding all the exponents together!

If you had something like $7^x \cdot 5^x$, you cannot combine them using this rule. The Product Rule only works when the bases are identical!