GMAT exponents strategies

When you multiply powers with the same base, you add the exponents: $a^m \times a^n = a^{m+n}$. When you divide powers with the same base, you subtract: $\frac{a^m}{a^n} = a^{m-n}$. When you raise a power to a power, you multiply the exponents: $(a^m)^n = a^{mn}$. Everything in exponents follows from these three rules. The key is matching the operation to the rule: multiplication of like bases adds exponents, division subtracts, and a power of a power multiplies.

The two cases look similar but use opposite rules. Same base, different exponents, and you multiply the bases: $2^3 \times 2^4 = 2^7$. Same exponent, different bases, and you multiply the bases and keep the exponent: $2^3 \times 5^3 = (2 \times 5)^3 = 10^3 = 1000$. Before applying any rule, check whether the bases match or the exponents match. If neither matches, rewrite the numbers so the bases do. For example, $4^3 \times 8^2$ becomes $(2^2)^3 \times (2^3)^2 = 2^6 \times 2^6 = 2^{12}$.

Simplify $\frac{2^5 \times 8^3}{4^4}$. Step 1: Rewrite to a common base of 2. The number 8 is $2^3$, so $8^3 = (2^3)^3 = 2^9$. The number 4 is $2^2$, so $4^4 = (2^2)^4 = 2^8$. Step 2: Substitute. The expression becomes $\frac{2^5 \times 2^9}{2^8}$. Step 3: Combine the numerator. Same base, multiplication, add exponents: $2^5 \times 2^9 = 2^{14}$. Now we have $\frac{2^{14}}{2^8}$. Step 4: Divide. Same base, division, subtract exponents: $2^{14-8} = 2^6 = 64$. The answer is 64.