Is the equation (3√4)^24/(√3)^6=(4/3)^5

Question

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False. The Quotient Property has not been applied correctly.

False. The Power of a Power Property has not been applied correctly.

True. The Power of a Power Property has been applied correctly.

True. The Quotient Property has been applied correctly.

Answer 1

To determine the validity of the equation \((3\sqrt{4})^{24}/(\sqrt{3})^{6}=(4/3)^{5}\), let's simplify both sides step by step.

Left Side: \[ \frac{(3\sqrt{4})^{24}}{(\sqrt{3})^{6}} \]

1. Simplify \( \sqrt{4} \): \[ \sqrt{4} = 2 \] So \( (3\sqrt{4})^{24} = (3 \cdot 2)^{24} = (6)^{24} \).

2. Now for the denominator: \[ (\sqrt{3})^{6} = (3^{1/2})^{6} = 3^{6/2} = 3^{3} = 27 \]

3. Now, rewrite the left side: \[ \frac{(6)^{24}}{27} = \frac{6^{24}}{3^{3}} = 6^{24} \cdot 3^{-3} \]

Right Side: \[ \left(\frac{4}{3}\right)^{5} = \frac{4^5}{3^5} \]

Comparison: To simplify further we can express \(6^{24}\): \[ 6^{24} = (2 \cdot 3)^{24} = 2^{24} \cdot 3^{24} \] Then substituting that in: \[ 6^{24} \cdot 3^{-3} = 2^{24} \cdot 3^{21} \]

Now let's simplify the right side: \[ \frac{4^5}{3^5} = \frac{(2^2)^5}{3^5} = \frac{2^{10}}{3^5} \]

Final Comparison: \[ 2^{24} \cdot 3^{21} \text{ (left side) } \quad \text{and } \quad \frac{2^{10}}{3^5} = 2^{10} \cdot 3^{-5} \text{ (right side)} \]

The powers of 2 and 3 on both sides are thus:

• Left side: \(2^{24}\) and \(3^{21}\)
• Right side: \(2^{10}\) and \(3^{-5}\).

Since these powers do not match, we can conclude:

The equation is False. The Power of a Power Property has not been applied correctly.

Thus, the correct response is: False. The Power of a Power Property has not been applied correctly.

Answer 2

it says its wrong

Answer 3

Let’s carefully analyze the equation again:

\[ \frac{(3\sqrt{4})^{24}}{(\sqrt{3})^{6}} = \left(\frac{4}{3}\right)^{5} \]

Step 1: Simplify the Left-Hand Side

First, simplify \((3\sqrt{4})^{24}\):

• \(\sqrt{4} = 2\), so: \[ 3\sqrt{4} = 3 \cdot 2 = 6 \] Thus: \[ (3\sqrt{4})^{24} = 6^{24} \]

Next, simplify \((\sqrt{3})^{6}\): \[ (\sqrt{3})^{6} = (3^{1/2})^{6} = 3^{6/2} = 3^{3} \]

Hence, the left-hand side becomes: \[ \frac{6^{24}}{3^{3}} = 6^{24} \cdot 3^{-3} \]

Now, express \(6\) in terms of its prime factors: \[ 6 = 2 \cdot 3, \text{ so } 6^{24} = (2 \cdot 3)^{24} = 2^{24} \cdot 3^{24} \]

The left side is: \[ \frac{6^{24}}{3^{3}} = \frac{2^{24} \cdot 3^{24}}{3^3} = 2^{24} \cdot 3^{21} \]

Step 2: Simplify the Right-Hand Side

Now simplify \(\left(\frac{4}{3}\right)^{5}\): \[ \left(\frac{4}{3}\right)^{5} = \frac{4^{5}}{3^{5}} = \frac{(2^2)^{5}}{3^{5}} = \frac{2^{10}}{3^{5}} \]

Step 3: Compare Both Sides

Now we have:

• Left Side: \(2^{24} \cdot 3^{21}\)
• Right Side: \(\frac{2^{10}}{3^{5}} = 2^{10} \cdot 3^{-5}\)

Step 4: Rewrite Right Side to Compare

We can rewrite the right side: \[ 2^{10} \cdot 3^{-5} = 2^{10} \cdot \frac{1}{3^{5}} = 2^{10} \cdot 3^{-5} \]

Conclusion

Since \(2^{24} \cdot 3^{21}\) on the left does not equal \(2^{10} \cdot 3^{-5}\) on the right, it is clear that the two sides are not equivalent.

Therefore, the statement is indeed False.

The correct justification for this is: False. The Power of a Power Property has not been applied correctly.

If the explanation in your original question suggests responding differently, please verify the original evaluation or specifications of the truth of the statement again with your source, as the simplifications lead to a different conclusion.

Answer 4

thats also wrong..

Answer 5

I apologize for the confusion earlier. Let's approach the problem step-by-step once again, ensuring we clarify every point meticulously.

The problem statement is:

\[ \frac{(3\sqrt{4})^{24}}{(\sqrt{3})^{6}}=\left(\frac{4}{3}\right)^{5} \]

Step 1: Simplify the Left Side

1. Calculate \( (3\sqrt{4})^{24} \):

   • First, simplify \(\sqrt{4}\): \[ \sqrt{4} = 2 \]
   • Therefore, \(3\sqrt{4} = 3 \cdot 2 = 6\).
   • Now raise \(6\) to the 24th power: \[ (3\sqrt{4})^{24} = 6^{24} \]

2. Calculate \( (\sqrt{3})^{6} \):

   • Using the properties of exponents: \[ (\sqrt{3})^{6} = (3^{1/2})^{6} = 3^{6/2} = 3^{3} \]

Putting it all together, we rewrite the left side: \[ \frac{(3\sqrt{4})^{24}}{(\sqrt{3})^{6}} = \frac{6^{24}}{3^{3}} \]

Step 2: Express \(6^{24}\) in Terms of Prime Factors

\[ 6^{24} = (2 \cdot 3)^{24} = 2^{24} \cdot 3^{24} \] Thus the left side becomes: \[ \frac{2^{24} \cdot 3^{24}}{3^{3}} = 2^{24} \cdot 3^{21} \]

Step 3: Simplify the Right Side

The right side is: \[ \left(\frac{4}{3}\right)^{5} = \frac{4^5}{3^5} \] Since \(4 = 2^2\), we can substitute: \[ \frac{(2^2)^5}{3^5} = \frac{2^{10}}{3^5} \]

Step 4: Set Up the Equation

Now we have:

• Left Side: \(2^{24} \cdot 3^{21}\)
• Right Side: \(2^{10} \cdot 3^{-5}\)

Step 5: Compare the Two Sides

We can equate \(2^{24} \cdot 3^{21}\) to \(2^{10} \cdot 3^{-5}\).

To determine if they are equal, compare the powers.

• For \(2\):
  • Left side: \(2^{24}\)
  • Right side: \(2^{10}\)

This shows \(24\) does not equal \(10\).

• For \(3\):
  • Left side: \(3^{21}\)
  • Right side: \(3^{-5}\)

This shows \(21\) does not equal \(-5\).

Conclusion

Therefore, since \(2^{24} \cdot 3^{21}\) does not equal \(2^{10} \cdot 3^{-5}\), the equation is indeed False.

The correct response is: False. The Power of a Power Property has not been applied correctly.

If there is still confusion or if the provided answer does not match the expected output, I would appreciate knowing the specific sources or instructions being followed, as my derivation and conclusions align consistently with mathematical principles. Thank you for your patience!