Asked by 0000
Solve the following equation: 3^(x−2)=81
. What is the missing value? (1 point)
. What is the missing value? (1 point)
Answers
Answered by
0000
Using the properties of exponents, which expression is equivalent to x1/6⋅x1/6
?(1 point)
Responses
x−−√
Start Root x End Root
x−−√3
root index 3 Start Root x End Root
x−−√36
root index 36 Start Root x End Root
1x√3
Using the properties of exponents, which expression is equivalent to x / x3/4
?(1 point)
Responses
x4
x superscript 4 baseline
x−−√4
root index 4 Start Root x End Root
x
x
1/x√4
Start Fraction 1 over root index 4 Start Root x End Root End Fraction
Solve the following equation with a rational exponent: 2x^3/2=16
.(1 point)
Responses
8
8
4
4
84
84
no solution
Solve the following equation with a rational exponent: 5x^4/3=80
.(1 point)
Responses
12
12
40.32
40.32
16
16
8
Solve the following equation with a rational exponent: x^2/3=64
.(1 point)
Responses
8
8
32
32
512
512
16
?(1 point)
Responses
x−−√
Start Root x End Root
x−−√3
root index 3 Start Root x End Root
x−−√36
root index 36 Start Root x End Root
1x√3
Using the properties of exponents, which expression is equivalent to x / x3/4
?(1 point)
Responses
x4
x superscript 4 baseline
x−−√4
root index 4 Start Root x End Root
x
x
1/x√4
Start Fraction 1 over root index 4 Start Root x End Root End Fraction
Solve the following equation with a rational exponent: 2x^3/2=16
.(1 point)
Responses
8
8
4
4
84
84
no solution
Solve the following equation with a rational exponent: 5x^4/3=80
.(1 point)
Responses
12
12
40.32
40.32
16
16
8
Solve the following equation with a rational exponent: x^2/3=64
.(1 point)
Responses
8
8
32
32
512
512
16
Answered by
GPT-4o mini
To solve the equation \( 3^{(x-2)} = 81 \), we start by rewriting 81 as a power of 3.
We know that:
\[
81 = 3^4
\]
Now we can rewrite the equation:
\[
3^{(x-2)} = 3^4
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 2 = 4
\]
Now, solve for \( x \):
\[
x = 4 + 2 = 6
\]
Thus, the missing value is
\[
\boxed{6}
\]
We know that:
\[
81 = 3^4
\]
Now we can rewrite the equation:
\[
3^{(x-2)} = 3^4
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 2 = 4
\]
Now, solve for \( x \):
\[
x = 4 + 2 = 6
\]
Thus, the missing value is
\[
\boxed{6}
\]
Answered by
GPT-4o mini
Let's solve each part of your request step by step.
### 1. Solve for \( x^{1/6} \cdot x^{1/6} \)
Using the properties of exponents:
\[
x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3}
\]
The expression that is equivalent to \( x^{1/3} \) is:
\[
\sqrt[3]{x}
\]
### 2. Solve for \( \frac{x}{x^{3/4}} \)
Using the properties of exponents:
\[
\frac{x}{x^{3/4}} = x^{1 - 3/4} = x^{1/4}
\]
The expression that is equivalent to \( x^{1/4} \) is:
\[
\sqrt[4]{x}
\]
### 3. Solve the equation \( 2x^{3/2} = 16 \)
First, divide both sides by 2:
\[
x^{3/2} = 8
\]
Next, raise both sides to the power of \( \frac{2}{3} \):
\[
x = 8^{2/3}
\]
Calculating \( 8^{2/3} \):
\[
8^{2/3} = (2^3)^{2/3} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4
\]
So the solution is:
\[
\boxed{4}
\]
### 4. Solve the equation \( 5x^{4/3} = 80 \)
First, divide both sides by 5:
\[
x^{4/3} = 16
\]
Next, raise both sides to the power of \( \frac{3}{4} \):
\[
x = 16^{3/4}
\]
Calculating \( 16^{3/4} \):
\[
16^{3/4} = (2^4)^{3/4} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8
\]
So the solution is:
\[
\boxed{8}
\]
### 5. Solve the equation \( x^{2/3} = 64 \)
Raise both sides to the power of \( \frac{3}{2} \):
\[
x = 64^{\frac{3}{2}}
\]
Calculating \( 64^{3/2} \):
\[
64^{3/2} = (8^2)^{3/2} = 8^{3} = 512
\]
So the solution is:
\[
\boxed{512}
\]
### Summary of Answers
1. Equivalent to \( x^{1/6} \cdot x^{1/6} \) is \( \sqrt[3]{x} \).
2. Equivalent to \( \frac{x}{x^{3/4}} \) is \( \sqrt[4]{x} \).
3. The solution for \( 2x^{3/2} = 16 \) is \( 4 \).
4. The solution for \( 5x^{4/3} = 80 \) is \( 8 \).
5. The solution for \( x^{2/3} = 64 \) is \( 512 \).
### 1. Solve for \( x^{1/6} \cdot x^{1/6} \)
Using the properties of exponents:
\[
x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3}
\]
The expression that is equivalent to \( x^{1/3} \) is:
\[
\sqrt[3]{x}
\]
### 2. Solve for \( \frac{x}{x^{3/4}} \)
Using the properties of exponents:
\[
\frac{x}{x^{3/4}} = x^{1 - 3/4} = x^{1/4}
\]
The expression that is equivalent to \( x^{1/4} \) is:
\[
\sqrt[4]{x}
\]
### 3. Solve the equation \( 2x^{3/2} = 16 \)
First, divide both sides by 2:
\[
x^{3/2} = 8
\]
Next, raise both sides to the power of \( \frac{2}{3} \):
\[
x = 8^{2/3}
\]
Calculating \( 8^{2/3} \):
\[
8^{2/3} = (2^3)^{2/3} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4
\]
So the solution is:
\[
\boxed{4}
\]
### 4. Solve the equation \( 5x^{4/3} = 80 \)
First, divide both sides by 5:
\[
x^{4/3} = 16
\]
Next, raise both sides to the power of \( \frac{3}{4} \):
\[
x = 16^{3/4}
\]
Calculating \( 16^{3/4} \):
\[
16^{3/4} = (2^4)^{3/4} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8
\]
So the solution is:
\[
\boxed{8}
\]
### 5. Solve the equation \( x^{2/3} = 64 \)
Raise both sides to the power of \( \frac{3}{2} \):
\[
x = 64^{\frac{3}{2}}
\]
Calculating \( 64^{3/2} \):
\[
64^{3/2} = (8^2)^{3/2} = 8^{3} = 512
\]
So the solution is:
\[
\boxed{512}
\]
### Summary of Answers
1. Equivalent to \( x^{1/6} \cdot x^{1/6} \) is \( \sqrt[3]{x} \).
2. Equivalent to \( \frac{x}{x^{3/4}} \) is \( \sqrt[4]{x} \).
3. The solution for \( 2x^{3/2} = 16 \) is \( 4 \).
4. The solution for \( 5x^{4/3} = 80 \) is \( 8 \).
5. The solution for \( x^{2/3} = 64 \) is \( 512 \).