Simplify the Expression: 7^5 × 7^(-6) Using Laws of Exponents

The product rule for exponents states that $a^m \cdot a^n = a^{m+n}$. This works because you're essentially multiplying the base by itself multiple times. For example, $7^2 \cdot 7^3 = (7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) = 7^5$.

A negative exponent means take the reciprocal and make the exponent positive. So $7^{-1} = \frac{1}{7^1} = \frac{1}{7}$. Think of it as 'flipping' the number to the denominator.

Yes, but be careful with the signs! Since we have $7^5 \cdot 7^{-6}$, we add the exponents: 5 + (-6) = 5 - 6 = -1. The key is recognizing that adding a negative number is the same as subtraction.

Use the product rule whenever you're multiplying terms with the same base. Look for expressions like $a^m \cdot a^n$. If the bases are different (like $7^2 \cdot 3^4$), you cannot use this rule.

Remember the simple rule: $a^{-n} = \frac{1}{a^n}$. A negative exponent always creates a fraction with 1 in the numerator. Practice with small numbers first, like $2^{-1} = \frac{1}{2}$.