The ratio $4^{3.5} : 2^5$ is the same as

### Concept & Equating Bases To simplify a ratio involving exponents with different bases, rewrite the terms so they share a common base. A power raised to another power means multiplying the exponents. $$ (x^a)^b = x^{a \times b} $$ ### Step-by-Step Solution * **Given:** The ratio $4^{3.5} : 2^5$. * **Identify a common base:** The number $4$ can be expressed as a base of $2$ ($4 = 2^2$). * **Rewrite the first term:** $4^{3.5} = (2^2)^{3.5}$ * **Apply the power rule:** Multiply the exponents $2 \times 3.5 = 7$. $4^{3.5} = 2^7$ * **Rewrite the ratio:** Substitute the new term back into the ratio. $2^7 : 2^5$ * **Simplify:** Divide both sides by the smaller power ($2^5$). $\frac{2^7}{2^5} : \frac{2^5}{2^5}$ $2^{7-5} : 1$ $2^2 : 1$ * **Final calculation:** Evaluate $2^2$. $4 : 1$ ### Exam Strategy & Shortcut Whenever you see bases like $2, 4, 8$, or $16$ in a ratio, immediately convert them all to base $2$. $(2^2)^{3.5}$ quickly becomes $2^7$. The ratio $2^7 / 2^5$ leaves $2^2$ in the numerator, which is $4$, making the answer $4 : 1$. ### Common Pitfall Failing to convert the bases first and attempting to estimate or falsely simplifying exponents across mismatched bases. ### Final Answer Therefore, the correct answer is **$4 : 1$**.