How can you simplify the exponential expression $\left(r^2 / s\right)^3 ?$
A. $\frac{r^{2+3}}{s^{1+3}}$
B. $\frac{r^{2 \cdot 3}}{s}$
C. $\frac{r^{2 \cdot 3}}{s^{1 \cdot 3}}$
D. $\frac{r^2+3}{s}$
Answer (1)
Apply the power of a quotient rule: ( r 2 / s ) 3 = s 3 ( r 2 ) 3 .
Apply the power of a power rule: ( r 2 ) 3 = r 2 ⋅ 3 = r 6 .
Simplify the denominator: s 3 = s 1 ⋅ 3 .
The simplified expression is s 1 ⋅ 3 r 2 ⋅ 3 .
Explanation
Understanding the Problem We are asked to simplify the exponential expression ( r 2 / s ) 3 . This involves applying the rules of exponents to both the numerator and the denominator inside the parentheses.
Applying the Power of a Quotient Rule We will use the power of a quotient rule, which states that ( b a ) n = b n a n . Applying this rule to our expression, we get: ( s r 2 ) 3 = s 3 ( r 2 ) 3
Applying the Power of a Power Rule Next, we apply the power of a power rule, which states that ( a m ) n = a m ⋅ n . Applying this rule to the numerator, we get: ( r 2 ) 3 = r 2 ⋅ 3 = r 6
Simplifying the Denominator Now, we consider the denominator. Since s = s 1 , we have s 3 = s 1 ⋅ 3 = s 3 . Thus, our expression becomes: s 3 r 6 This is equivalent to s 1 ⋅ 3 r 2 ⋅ 3 .
Final Answer Therefore, the simplified expression is s 3 r 6 , which corresponds to the option s 1 ⋅ 3 r 2 ⋅ 3 .
Examples
Exponential expressions are used in various fields, such as finance and physics. For example, when calculating compound interest, the formula involves raising the interest rate plus one to the power of the number of compounding periods. Similarly, in physics, exponential decay describes the decrease in the amount of a radioactive substance over time. Understanding how to simplify exponential expressions is crucial for solving problems in these areas.
