Let's solve each question one by one.
Question 1: We need to rewrite \( 3^{-1/2} \) in radical form using the Negative Exponent Rule.
Using the Negative Exponent Rule: \[ a^{-n} = \frac{1}{a^n} \] we can write: \[ 3^{-1/2} = \frac{1}{3^{1/2}} = \frac{1}{\sqrt{3}} \]
Answer for Question 1: \[ 3^{-1/2} = \frac{1}{\sqrt{3}} \]
Question 2: We will apply the Quotient Property of exponents: \[ \frac{a^m}{a^n} = a^{m-n} \] Given expression is: \[ \frac{9^{3/5}}{9^{1/5}} \] So: \[ 9^{3/5 - 1/5} = 9^{2/5} \]
Answer for Question 2: The simplified exponent is: \[ \frac{2}{5} \]
Question 3: We need to solve for \( x \) in the equation: \[ x^{1/5} = 6 \] To solve for \( x \), we raise both sides to the power of 5: \[ x = 6^5 \] Calculating \( 6^5 \): \[ 6^5 = 7776 \]
Answer for Question 3: \[ x = 7776 \]
Question 4: We will solve the equation: \[ 2^{x+2} = 4^2 \] First, rewrite \( 4 \) as \( 2^2 \): \[ 4^2 = (2^2)^2 = 2^{2 \cdot 2} = 2^4 \] Now, set the exponents equal to each other: \[ x + 2 = 4 \] Solving for \( x \): \[ x = 4 - 2 = 2 \]
Answer for Question 4: \[ x = 2 \]
Question 5: We will solve the equation: \[ 3^{x-2} = 81 \] First, rewrite \( 81 \) as a power of \( 3 \): \[ 81 = 3^4 \] Setting the exponents equal: \[ x - 2 = 4 \] Now, solve for \( x \): \[ x = 4 + 2 = 6 \]
Answer for Question 5: \[ x = 6 \]
To summarize the answers:
- \( 3^{-1/2} = \frac{1}{\sqrt{3}} \)
- The simplified exponent is \( \frac{2}{5} \)
- \( x = 7776 \)
- \( x = 2 \)
- \( x = 6 \)
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