Solve the following equation: 3^(x−2)=81

Question

. What is the missing value? (1 point)

Answer 1

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

\[ \boxed{6} \]

Answer 2

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

Answer 3

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 \( 512 \).