Question
Using the exponent properties, which of the following expressions is equivalent to (√64)^2/3⋅64^0
?(1 point)
Responses
64−13
64 superscript negative Start Fraction 1 over 3 End Fraction baseline
14
Start Fraction 1 over 4 End Fraction
4
4
1
?(1 point)
Responses
64−13
64 superscript negative Start Fraction 1 over 3 End Fraction baseline
14
Start Fraction 1 over 4 End Fraction
4
4
1
Answers
Answer
Which of the following expressions is equivalent to 16^1/4 devided by √49
? Use the properties of exponents to help determine the answer.(1 point)
Responses
47
Start Fraction 4 over 7 End Fraction
2
2
12
Start Fraction 1 over 2 End Fraction
27
? Use the properties of exponents to help determine the answer.(1 point)
Responses
47
Start Fraction 4 over 7 End Fraction
2
2
12
Start Fraction 1 over 2 End Fraction
27
Answer
Is the equation (3√4)^24 devided by (√3)^6=(4/3)^5
true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses
True. The Power of a Power Property has been applied correctly.
True. The Power of a Power Property has been applied correctly. - incorrect
False. The Quotient Property has not been applied correctly.
False. The Quotient Property has not been applied correctly. - no response given
False. The Power of a Power Property has not been applied correctly.
False. The Power of a Power Property has not been applied correctly. - no response given
True. The Quotient Property has been applied correctly.
true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses
True. The Power of a Power Property has been applied correctly.
True. The Power of a Power Property has been applied correctly. - incorrect
False. The Quotient Property has not been applied correctly.
False. The Quotient Property has not been applied correctly. - no response given
False. The Power of a Power Property has not been applied correctly.
False. The Power of a Power Property has not been applied correctly. - no response given
True. The Quotient Property has been applied correctly.
Answer
Is the equation 6^1/2⋅(√6)^3=36
true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses
True, according to the Power of a Power Property.
True, according to the Power of a Power Property.
False, because the exponents should be divided.
False, because the exponents should be divided.
True, according to the Product Property.
True, according to the Product Property.
False, because the exponents should be multiplied.
true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses
True, according to the Power of a Power Property.
True, according to the Power of a Power Property.
False, because the exponents should be divided.
False, because the exponents should be divided.
True, according to the Product Property.
True, according to the Product Property.
False, because the exponents should be multiplied.
Answered by
GPT-4o mini
To simplify the expression \((\sqrt{64})^{2/3} \cdot 64^0\), let's break it down step by step.
1. Simplify \(\sqrt{64}\):
\[
\sqrt{64} = 8
\]
2. Raise \(8\) to the power \(\frac{2}{3}\):
\[
(8)^{2/3} = (2^3)^{2/3} = 2^{3 \cdot (2/3)} = 2^2 = 4
\]
3. Evaluate \(64^0\):
\[
64^0 = 1
\]
4. Now substitute back into the expression:
\[
4 \cdot 1 = 4
\]
The simplified expression is \(4\).
So, the equivalent expression is:
\[
\boxed{4}
\]
1. Simplify \(\sqrt{64}\):
\[
\sqrt{64} = 8
\]
2. Raise \(8\) to the power \(\frac{2}{3}\):
\[
(8)^{2/3} = (2^3)^{2/3} = 2^{3 \cdot (2/3)} = 2^2 = 4
\]
3. Evaluate \(64^0\):
\[
64^0 = 1
\]
4. Now substitute back into the expression:
\[
4 \cdot 1 = 4
\]
The simplified expression is \(4\).
So, the equivalent expression is:
\[
\boxed{4}
\]
Answered by
GPT-4o mini
To simplify the expression \( \frac{16^{1/4}}{\sqrt{49}} \), we will perform the calculations step by step.
1. Simplify \( 16^{1/4} \):
\[
16 = 2^4 \quad \Rightarrow \quad 16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot (1/4)} = 2^1 = 2
\]
2. Simplify \( \sqrt{49} \):
\[
\sqrt{49} = 7
\]
3. Now we can substitute back into the expression:
\[
\frac{16^{1/4}}{\sqrt{49}} = \frac{2}{7}
\]
Now, the simplified expression is:
\[
\frac{2}{7}
\]
Therefore, the correct equivalent expression is:
\[
\boxed{\frac{2}{7}}
\]
1. Simplify \( 16^{1/4} \):
\[
16 = 2^4 \quad \Rightarrow \quad 16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot (1/4)} = 2^1 = 2
\]
2. Simplify \( \sqrt{49} \):
\[
\sqrt{49} = 7
\]
3. Now we can substitute back into the expression:
\[
\frac{16^{1/4}}{\sqrt{49}} = \frac{2}{7}
\]
Now, the simplified expression is:
\[
\frac{2}{7}
\]
Therefore, the correct equivalent expression is:
\[
\boxed{\frac{2}{7}}
\]
Answered by
GPT-4o mini
To determine whether the equation
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \left(\frac{4}{3}\right)^{5}
\]
is true or false, let’s simplify both sides step by step.
### Left Side: \(\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6}\)
1. **Simplify \((3\sqrt{4})^{24}\)**:
The square root of \(4\) is \(2\):
\[
3\sqrt{4} = 3 \times 2 = 6
\]
Thus:
\[
(3\sqrt{4})^{24} = 6^{24} = (6^2)^{12} = 36^{12}
\]
2. **Simplify \((\sqrt{3})^6\)**:
\[
(\sqrt{3})^6 = (3^{1/2})^6 = 3^{6/2} = 3^3 = 27
\]
3. **Combine Left Side**:
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \frac{6^{24}}{27}
\]
### Right Side: \(\left(\frac{4}{3}\right)^{5}\)
1. **Calculate \(\left(\frac{4}{3}\right)^{5}\)**:
\[
\left(\frac{4}{3}\right)^{5} = \frac{4^5}{3^5}
\]
2. **Calculate \(4^5\) and \(3^5\)**:
\[
4^5 = 1024 \quad \text{and} \quad 3^5 = 243
\]
Thus:
\[
\left(\frac{4}{3}\right)^{5} = \frac{1024}{243}
\]
### Compare Both Sides
Now we compare:
1. Simplify \(6^{24}\):
\[
6^{24} = (2 \cdot 3)^{24} = 2^{24} \cdot 3^{24}
\]
Thus:
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \frac{2^{24} \cdot 3^{24}}{27} = \frac{2^{24} \cdot 3^{24}}{3^3} = 2^{24} \cdot 3^{21}
\]
The left side simplifies to \(2^{24} \cdot 3^{21}\).
2. Since \(\left(\frac{4}{3}\right)^{5} = \frac{1024}{243}\), further analysis shows the powers of \(2\) and \(3\) are not equivalent with the left side \(2^{24} \cdot 3^{21}\).
### Conclusion:
Since both sides yield different results, the equation is **false**.
**Final Answer:**
\[
\text{False. The Power of a Power Property has not been applied correctly.}
\]
Thus, the correct justification is the last option:
**False. The Power of a Power Property has not been applied correctly.**
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \left(\frac{4}{3}\right)^{5}
\]
is true or false, let’s simplify both sides step by step.
### Left Side: \(\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6}\)
1. **Simplify \((3\sqrt{4})^{24}\)**:
The square root of \(4\) is \(2\):
\[
3\sqrt{4} = 3 \times 2 = 6
\]
Thus:
\[
(3\sqrt{4})^{24} = 6^{24} = (6^2)^{12} = 36^{12}
\]
2. **Simplify \((\sqrt{3})^6\)**:
\[
(\sqrt{3})^6 = (3^{1/2})^6 = 3^{6/2} = 3^3 = 27
\]
3. **Combine Left Side**:
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \frac{6^{24}}{27}
\]
### Right Side: \(\left(\frac{4}{3}\right)^{5}\)
1. **Calculate \(\left(\frac{4}{3}\right)^{5}\)**:
\[
\left(\frac{4}{3}\right)^{5} = \frac{4^5}{3^5}
\]
2. **Calculate \(4^5\) and \(3^5\)**:
\[
4^5 = 1024 \quad \text{and} \quad 3^5 = 243
\]
Thus:
\[
\left(\frac{4}{3}\right)^{5} = \frac{1024}{243}
\]
### Compare Both Sides
Now we compare:
1. Simplify \(6^{24}\):
\[
6^{24} = (2 \cdot 3)^{24} = 2^{24} \cdot 3^{24}
\]
Thus:
\[
\frac{(3\sqrt{4})^{24}}{(\sqrt{3})^6} = \frac{2^{24} \cdot 3^{24}}{27} = \frac{2^{24} \cdot 3^{24}}{3^3} = 2^{24} \cdot 3^{21}
\]
The left side simplifies to \(2^{24} \cdot 3^{21}\).
2. Since \(\left(\frac{4}{3}\right)^{5} = \frac{1024}{243}\), further analysis shows the powers of \(2\) and \(3\) are not equivalent with the left side \(2^{24} \cdot 3^{21}\).
### Conclusion:
Since both sides yield different results, the equation is **false**.
**Final Answer:**
\[
\text{False. The Power of a Power Property has not been applied correctly.}
\]
Thus, the correct justification is the last option:
**False. The Power of a Power Property has not been applied correctly.**
Answered by
GPT-4o mini
To determine if the equation
\[
6^{1/2} \cdot (\sqrt{6})^3 = 36
\]
is true or false, let's simplify the left side and see if it equals \(36\).
### Simplifying the Left Side:
1. Start with \(6^{1/2}\):
\[
6^{1/2} = \sqrt{6}
\]
2. Now consider \((\sqrt{6})^3\):
\[
\sqrt{6} = 6^{1/2} \quad \Rightarrow \quad (\sqrt{6})^3 = (6^{1/2})^3 = 6^{(1/2) \cdot 3} = 6^{3/2}
\]
3. Now combine the two parts:
\[
6^{1/2} \cdot 6^{3/2} = 6^{(1/2 + 3/2)} = 6^{4/2} = 6^2 = 36
\]
### Conclusion:
Since we have simplified the left side to \(36\), the equation is indeed true.
### Correct Response:
**True, according to the Product Property.**
\[
6^{1/2} \cdot (\sqrt{6})^3 = 36
\]
is true or false, let's simplify the left side and see if it equals \(36\).
### Simplifying the Left Side:
1. Start with \(6^{1/2}\):
\[
6^{1/2} = \sqrt{6}
\]
2. Now consider \((\sqrt{6})^3\):
\[
\sqrt{6} = 6^{1/2} \quad \Rightarrow \quad (\sqrt{6})^3 = (6^{1/2})^3 = 6^{(1/2) \cdot 3} = 6^{3/2}
\]
3. Now combine the two parts:
\[
6^{1/2} \cdot 6^{3/2} = 6^{(1/2 + 3/2)} = 6^{4/2} = 6^2 = 36
\]
### Conclusion:
Since we have simplified the left side to \(36\), the equation is indeed true.
### Correct Response:
**True, according to the Product Property.**