Question

If r and s are positive real numbers, which expression is equivalent to startfraction r superscript startfraction 2 over 3 endfraction baseline s superscript startfraction 1 over 2 endfraction baseline over r s endfraction?

a) r^2/3s^1/2
b) r^1/2s^2/3
c) r^2/3s^2/3
d) r^1/3s^1/2

Answer 1

The simplified expression of \(\frac{r^{\frac{2}{3}} s^{\frac{1}{2}}}{rs}\) is \(r^{\frac{1}{3}}s^{\frac{1}{2}}\), after subtracting the exponents due to division. However, this answer is not listed in the provided options, suggesting there may be an error in the question or the answer choices.

Step-by-step explanation:

To find the expression equivalent to \(\frac{r^{\frac{2}{3}} s^{\frac{1}{2}}}{rs}\), we need to simplify the fraction by dividing the exponents. We know that when we divide terms with the same base, we subtract exponents, according to the rules of exponents. Applying this rule:

• For r: \(r^{\frac{2}{3}}\) divided by r is the same as \(r^{\frac{2}{3} - 1} = r^{-\frac{1}{3}}\)
• For s: \(s^{\frac{1}{2}}\) divided by s is the same as \(s^{\frac{1}{2} - 1} = s^{-\frac{1}{2}}\)

The negative exponents indicate that the factors are on the wrong side of the fraction. To express them with positive exponents, we flip them to the other side:

• \(r^{-\frac{1}{3}}\) becomes \(r^{\frac{1}{3}}\) on the denominator.
• \(s^{-\frac{1}{2}}\) becomes \(s^{\frac{1}{2}}\) on the denominator.

Therefore, the simplified form of the expression is \(r^{\frac{1}{3}}s^{\frac{1}{2}}\), which is not one of the given options. There might be a mistake in the question or the answer choices provided.