Question

. Use the Negative Exponent Rule to generate an equivalent expression to 3−12
. Write your answer in radical form.(1 point)
3−12 =


Question 2
2. Use the Quotient Property to generate an equivalent expression to 935915
. What is the simplified exponent in fraction form?



(1 point)
The simplified exponent is



Question 3
3. What is the value of x
in x15=6?
(1 point)
x=


Question 4
4. Solve the following equation 2x+2=42
. What is the value of x
?(1 point)
x=


Question 5
5. Solve the following equation: 3x−2=81.
What is the value of x
? (1 point)
x =

Answer 1

Let’s address your questions one by one:

### Question 1
Using the Negative Exponent Rule, we know that \( a^{-n} = \frac{1}{a^n} \).

So,
\[
3^{-12} = \frac{1}{3^{12}}
\]

To write it in radical form, we can express the exponent as a root:
\[
3^{-12} = \frac{1}{3^{12}} = \frac{1}{\sqrt[1/12]{3^{12}}} = \frac{1}{(\sqrt[12]{3})^{12}} = \frac{1}{\sqrt[12]{3^{12}}}
\]

Thus, the equivalent expression is:
\[
3^{-12} = \frac{1}{\sqrt[12]{3^{12}}}
\]

### Question 2
Using the Quotient Property, which states that \( \frac{a^m}{a^n} = a^{m-n} \), let's simplify \( \frac{9^3}{9^5} \):

Calculating the exponent:
\[
3 - 5 = -2
\]

So, the simplified exponent in fraction form is:
\[
-2 = -\frac{2}{1}
\]

### Question 3
To solve for \( x \) in the equation \( x^{15} = 6 \), we can take the 15th root of both sides:

\[
x = 6^{\frac{1}{15}}
\]

### Question 4
To solve the equation \( 2^{x+2} = 42 \), we can rewrite it in logarithm form:

\[
x + 2 = \log_2(42)
\]
Now subtract 2 from both sides to solve for \( x \):
\[
x = \log_2(42) - 2
\]

### Question 5
To solve the equation \( 3^{x-2} = 81 \), we first rewrite \( 81 \) as a power of \( 3 \):

\[
3^{x-2} = 3^4
\]

Setting the exponents equal, we have:
\[
x - 2 = 4
\]
Now, adding 2 to both sides:
\[
x = 6
\]

### Summary of Answers
1. \( 3^{-12} = \frac{1}{3^{12}} \)
2. The simplified exponent is \( -\frac{2}{1} \)
3. \( x = 6^{\frac{1}{15}} \)
4. \( x = \log_2(42) - 2 \)
5. \( x = 6 \)