Question
1 of 11 of 1 Items
Question 1
1. According to the exponent rules, what is the rational exponent in the equivalent expression to 5–√3
? Enter your response in fraction form. (Hint: what should go in the yellow box)(1 point)
The rational exponent is
Question 2
2. Use the properties of exponents to generate an equivalent expression for 8√32√
(1 point)
8√32√=
Question 3
3. Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for 49√7−3
. What will the rational exponent be for the new expression? (Hint: what should go in the yellow box)(1 point)
The rational exponent for the new expression is
Question 4
4. Consider the equation 43⋅(64−−√3)2=46
. Is this equation true or false? (1 point)
Responses
True
True
False
False
Question 5
5. Consider the equation: (25−−√)4⋅56=58
. Is this equation true or false?(1 point)
Responses
True
True
False
False
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Answers
GPT-4o mini
Let's go through each question step by step.
**Question 1:**
1. The expression \( 5^{-\sqrt{3}} \) can be converted to a rational exponent. Remember that a negative exponent means to take the reciprocal, and the root can be expressed as a fraction. Thus,
\[
-\sqrt{3} = -\frac{3}{2}
\]
The rational exponent is \(-\frac{3}{2}\).
**Question 2:**
2. To simplify \( 8\sqrt{32} \), start by rewriting both expressions:
- \( 8 = 2^3 \)
- \( \sqrt{32} = \sqrt{2^5} = 2^{5/2} \)
Combining these gives:
\[
8\sqrt{32} = 2^3 \cdot 2^{5/2} = 2^{3 + 5/2} = 2^{\frac{6}{2} + \frac{5}{2}} = 2^{\frac{11}{2}}
\]
Thus, \( 8\sqrt{32} = 2^{\frac{11}{2}} \).
**Question 3:**
3. The expression \( 49\sqrt{7^{-3}} \) can be rewritten.
\[
49 = 7^2 \quad \text{and} \quad \sqrt{7^{-3}} = 7^{-3/2}
\]
Therefore,
\[
49\sqrt{7^{-3}} = 7^2 \cdot 7^{-3/2} = 7^{2 + (-3/2)} = 7^{\frac{4}{2} - \frac{3}{2}} = 7^{\frac{1}{2}}
\]
The rational exponent for the new expression is \( \frac{1}{2} \).
**Question 4:**
4. Simplifying the left side of the equation:
\[
43 \cdot (64^{3/2}) = 43 \cdot (8^3) = 43 \cdot 512 = 43 \cdot 2^9 = 2^{4} \cdot 2^{9} = 2^{13};
\]
The right side is:
\[
46 = 2^{6}
\]
Since \( 2^{13} \) does not equal \( 2^{6} \), the equation is **False**.
**Question 5:**
5. Simplifying the left side of the equation:
\[
(25^{1/2})^4 = (5^2)^{4} = 5^{8};
\]
The right side is:
\[
56 = 5^8
\]
Since both sides are equal, the equation is **True**.
To summarize:
1. The rational exponent is \(-\frac{3}{2}\).
2. \(8\sqrt{32} = 2^{\frac{11}{2}}\).
3. The rational exponent for the new expression is \(\frac{1}{2}\).
4. The equation is False.
5. The equation is True.
**Question 1:**
1. The expression \( 5^{-\sqrt{3}} \) can be converted to a rational exponent. Remember that a negative exponent means to take the reciprocal, and the root can be expressed as a fraction. Thus,
\[
-\sqrt{3} = -\frac{3}{2}
\]
The rational exponent is \(-\frac{3}{2}\).
**Question 2:**
2. To simplify \( 8\sqrt{32} \), start by rewriting both expressions:
- \( 8 = 2^3 \)
- \( \sqrt{32} = \sqrt{2^5} = 2^{5/2} \)
Combining these gives:
\[
8\sqrt{32} = 2^3 \cdot 2^{5/2} = 2^{3 + 5/2} = 2^{\frac{6}{2} + \frac{5}{2}} = 2^{\frac{11}{2}}
\]
Thus, \( 8\sqrt{32} = 2^{\frac{11}{2}} \).
**Question 3:**
3. The expression \( 49\sqrt{7^{-3}} \) can be rewritten.
\[
49 = 7^2 \quad \text{and} \quad \sqrt{7^{-3}} = 7^{-3/2}
\]
Therefore,
\[
49\sqrt{7^{-3}} = 7^2 \cdot 7^{-3/2} = 7^{2 + (-3/2)} = 7^{\frac{4}{2} - \frac{3}{2}} = 7^{\frac{1}{2}}
\]
The rational exponent for the new expression is \( \frac{1}{2} \).
**Question 4:**
4. Simplifying the left side of the equation:
\[
43 \cdot (64^{3/2}) = 43 \cdot (8^3) = 43 \cdot 512 = 43 \cdot 2^9 = 2^{4} \cdot 2^{9} = 2^{13};
\]
The right side is:
\[
46 = 2^{6}
\]
Since \( 2^{13} \) does not equal \( 2^{6} \), the equation is **False**.
**Question 5:**
5. Simplifying the left side of the equation:
\[
(25^{1/2})^4 = (5^2)^{4} = 5^{8};
\]
The right side is:
\[
56 = 5^8
\]
Since both sides are equal, the equation is **True**.
To summarize:
1. The rational exponent is \(-\frac{3}{2}\).
2. \(8\sqrt{32} = 2^{\frac{11}{2}}\).
3. The rational exponent for the new expression is \(\frac{1}{2}\).
4. The equation is False.
5. The equation is True.