Question

Is the equation ((root(4, 3)) ^ 24)/((sqrt(3)) ^ 6) = (4/3) ^ 5 true or false? Select the response that correctly answers the question and appropriate justification. (1 point) True. The Power of a Power Property has been applied correctly. True. The Quotient Property has been applied correctly. False. The Power of a Power Property has not been applied correctly. False. The Quotient Property has not been applied correctly.

Answer 1

To evaluate the equation \(\frac{(\sqrt[4]{3})^{24}}{(\sqrt{3})^6} = \left(\frac{4}{3}\right)^5\), let's first simplify the left-hand side.

1. **Evaluate \(\sqrt[4]{3}\):**
\(\sqrt[4]{3} = 3^{1/4}\)

So, \((\sqrt[4]{3})^{24} = (3^{1/4})^{24} = 3^{(1/4) \cdot 24} = 3^6\).

2. **Evaluate \(\sqrt{3}\):**
\(\sqrt{3} = 3^{1/2}\)

So, \((\sqrt{3})^6 = (3^{1/2})^6 = 3^{(1/2) \cdot 6} = 3^3\).

3. **Now substitute these values back into the left-hand side:**
\[
\frac{3^6}{3^3} = 3^{6-3} = 3^3
\]

The left-hand side simplifies to \(3^3\).

4. **Now evaluate the right-hand side:**
\[
\left(\frac{4}{3}\right)^5 = \frac{4^5}{3^5}
\]

5. **Comparing both sides:**
We have:
- Left-hand side: \(3^3\)
- Right-hand side: \(\frac{4^5}{3^5}\)

To compare these, let's also express \(3^3\) using a common denominator:
\[
3^3 = \frac{27}{1} = \frac{27 \cdot 3^5}{3^5} = \frac{27 \cdot 3^5}{3^5} = \frac{27}{3^5}
\]

6. **Now we compare \(27\) and \(4^5\):**
We know \(4^5 = 1024\), so \(\frac{27}{3^5}\) is not equal to \(\frac{1024}{3^5}\).

Thus, since \(3^3 \neq \left(\frac{4}{3}\right)^5\), the original equation is false.

**Final Answer:**
False. The Power of a Power Property has not been applied correctly.