Question
Use the laws of exponents to solve for x in the equation
x
4
4
1
=4
4
3
.
A. −
2
1
B.
2
1
C. 2 D. 16
x
4
4
1
=4
4
3
.
A. −
2
1
B.
2
1
C. 2 D. 16
Answers
GPT-4o mini
To solve for \( x \) in the equation
\[
x^{4/4} = 4^{3/4},
\]
we can simplify the left side:
\[
x^{4/4} = x^1 = x.
\]
So the equation simplifies to
\[
x = 4^{3/4}.
\]
Now, let's simplify \( 4^{3/4} \):
First, we can express \( 4 \) as \( 2^2 \):
\[
4^{3/4} = (2^2)^{3/4}.
\]
Using the laws of exponents, we multiply the exponents:
\[
(2^2)^{3/4} = 2^{2 \cdot (3/4)} = 2^{6/4} = 2^{3/2}.
\]
We can rewrite \( 2^{3/2} \) as:
\[
2^{3/2} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2}.
\]
Alternatively, if we want to express \( 4^{3/4} \) as a decimal or an integer, we calculate \( 4^{3/4} = 2^{3} = 8 \) because \( 4^{1/4} = 2^{1/2} = \sqrt{2} \) and thus \( 4^{3/4} = \sqrt{2}^3 = 8^{3/2} \).
Now let's check which of the options matches \( x = 2\sqrt{2} \):
- A. \( -2 \)
- B. \( 2 \)
- C. \( 2 \)
- D. \( 16 \)
None of the options explicitly match \( 2\sqrt{2} \). Thus, let's approximate \( 2\sqrt{2} \):
Since \( \sqrt{2} \approx 1.414 \),
\[
2\sqrt{2} \approx 2 \times 1.414 \approx 2.828.
\]
However, if \( \sqrt{2} \) is included, check for other integer or rational representations.
Given all computations, we find \( x \approx 2.828 \).
Ultimately, if looking at typical integer solutions including integer approximations, conclude what the allowable options depict as though interpreted within \( 2 \cdot \sqrt{2} \), clearly delivering value around height known in various contexts but one should verify presenting forms within limits of rational approximation.
Hence, it appears that none fits \( x \) described within main rational or integer outcomes conclusively.
Would proceed back reflecting perhaps numerical standards near full exponents demonstrates discrepancies within provided set.
\[
x^{4/4} = 4^{3/4},
\]
we can simplify the left side:
\[
x^{4/4} = x^1 = x.
\]
So the equation simplifies to
\[
x = 4^{3/4}.
\]
Now, let's simplify \( 4^{3/4} \):
First, we can express \( 4 \) as \( 2^2 \):
\[
4^{3/4} = (2^2)^{3/4}.
\]
Using the laws of exponents, we multiply the exponents:
\[
(2^2)^{3/4} = 2^{2 \cdot (3/4)} = 2^{6/4} = 2^{3/2}.
\]
We can rewrite \( 2^{3/2} \) as:
\[
2^{3/2} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2}.
\]
Alternatively, if we want to express \( 4^{3/4} \) as a decimal or an integer, we calculate \( 4^{3/4} = 2^{3} = 8 \) because \( 4^{1/4} = 2^{1/2} = \sqrt{2} \) and thus \( 4^{3/4} = \sqrt{2}^3 = 8^{3/2} \).
Now let's check which of the options matches \( x = 2\sqrt{2} \):
- A. \( -2 \)
- B. \( 2 \)
- C. \( 2 \)
- D. \( 16 \)
None of the options explicitly match \( 2\sqrt{2} \). Thus, let's approximate \( 2\sqrt{2} \):
Since \( \sqrt{2} \approx 1.414 \),
\[
2\sqrt{2} \approx 2 \times 1.414 \approx 2.828.
\]
However, if \( \sqrt{2} \) is included, check for other integer or rational representations.
Given all computations, we find \( x \approx 2.828 \).
Ultimately, if looking at typical integer solutions including integer approximations, conclude what the allowable options depict as though interpreted within \( 2 \cdot \sqrt{2} \), clearly delivering value around height known in various contexts but one should verify presenting forms within limits of rational approximation.
Hence, it appears that none fits \( x \) described within main rational or integer outcomes conclusively.
Would proceed back reflecting perhaps numerical standards near full exponents demonstrates discrepancies within provided set.