Question

Use the Negative Exponent Rule to generate an equivalent expression to 3^−1/2
.(1 point)

Answer 1

Use the Quotient Property to generate an equivalent expression to 9 3/5 / 9 1/5
. What is the simplified exponent?(1 point)
$$

Answer 2

What is the missing value in x^1/5=6
?(1 point)

Answer 3

Solve the following equation: 2^(x+2)=42
. What is the missing value?(1 point)

Answer 4

The Negative Exponent Rule states that \( a^{-n} = \frac{1}{a^n} \).

Using this rule for the expression \( 3^{-1/2} \), we can rewrite it as:

\[
3^{-1/2} = \frac{1}{3^{1/2}}
\]

The expression \( 3^{1/2} \) is the same as \( \sqrt{3} \). Therefore, we can further simplify the expression:

\[
3^{-1/2} = \frac{1}{\sqrt{3}}
\]

Thus, an equivalent expression to \( 3^{-1/2} \) is:

\[
\frac{1}{\sqrt{3}}
\]

Answer 5

To use the Quotient Property of exponents, we first need to express the mixed numbers as improper fractions.

1. **Convert the mixed numbers:**
- \( 9 \frac{3}{5} = 9 + \frac{3}{5} = \frac{45}{5} + \frac{3}{5} = \frac{48}{5} \)
- \( 9 \frac{1}{5} = 9 + \frac{1}{5} = \frac{45}{5} + \frac{1}{5} = \frac{46}{5} \)

Now, we can rewrite the original expression using these improper fractions:

\[
\frac{9 \frac{3}{5}}{9 \frac{1}{5}} = \frac{\frac{48}{5}}{\frac{46}{5}}
\]

2. **Applying the Quotient Property:**
The Quotient Property states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, since we can simplify the fractions directly:

\[
\frac{\frac{48}{5}}{\frac{46}{5}} = \frac{48}{46} = \frac{24}{23}
\]

3. **Using the exponents:**
To relate this back to exponents, note that \( 9 \) can be expressed as \( 9^1 \). Therefore, we can write the expression in exponential form before simplifying:

The original expression can be represented in terms of a base:

\[
\frac{9^{3/5}}{9^{1/5}} = 9^{\frac{3}{5} - \frac{1}{5}}
\]

4. **Simplifying the exponent:**

Now, calculate the exponent:

\[
\frac{3}{5} - \frac{1}{5} = \frac{2}{5}
\]

Thus, we have:

\[
9^{\frac{2}{5}}
\]

The simplified exponent is:

\[
\frac{2}{5}
\]

So, the equivalent expression in simplified exponent form is:

\[
9^{\frac{2}{5}}
\] with the simplified exponent being \( \frac{2}{5} \).

Answer 6

To solve for the missing value in \( x^{1/5} = 6 \), we can isolate \( x \) by raising both sides of the equation to the power of 5 (which is the inverse of taking the fifth root):

\[
(x^{1/5})^5 = 6^5
\]

This simplifies to:

\[
x = 6^5
\]

Now, we can compute \( 6^5 \):

\[
6^5 = 6 \times 6 \times 6 \times 6 \times 6
\]

Calculating this step by step:

1. \( 6 \times 6 = 36 \)
2. \( 36 \times 6 = 216 \)
3. \( 216 \times 6 = 1296 \)
4. \( 1296 \times 6 = 7776 \)

Thus, \( 6^5 = 7776 \).

Therefore, the missing value is:

\[
x = 7776
\]

Answer 7

To solve the equation \( 2^{(x+2)} = 42 \), we will use logarithms. Here are the steps to find the value of \( x \):

1. **Take the logarithm of both sides:**

You can use the natural logarithm (ln) or the common logarithm (log). For this example, let's use the natural logarithm:

\[
\ln(2^{(x+2)}) = \ln(42)
\]

2. **Apply the power rule of logarithms:**

The power rule states that \(\ln(a^b) = b \cdot \ln(a)\). Thus, we can rewrite our equation as:

\[
(x+2) \cdot \ln(2) = \ln(42)
\]

3. **Solve for \( x+2 \):**

Divide both sides by \(\ln(2)\):

\[
x + 2 = \frac{\ln(42)}{\ln(2)}
\]

4. **Isolate \( x \):**

Subtract 2 from both sides:

\[
x = \frac{\ln(42)}{\ln(2)} - 2
\]

5. **Calculate the value:**

Now, we can calculate the numerical value using a calculator. First, calculate \(\ln(42)\) and \(\ln(2)\):

- \(\ln(42) \approx 3.7377\)
- \(\ln(2) \approx 0.6931\)

Now plug these values into the equation:

\[
x \approx \frac{3.7377}{0.6931} - 2
\]

Calculating that:

\[
x \approx 5.390 - 2 \approx 3.390
\]

So the missing value is approximately:

\[
\boxed{3.39}
\]