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

Question

.(1 point)

Answer 1

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 2

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

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 4

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

Answer 5

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 6

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

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:

\[ \boxed{3.39} \]