Let's go through each question step by step.
Question 1:
- The expression \( 5^{-\sqrt{3}} \) can be converted to a rational exponent. Remember that a negative exponent means to take the reciprocal, and the root can be expressed as a fraction. Thus, \[ -\sqrt{3} = -\frac{3}{2} \] The rational exponent is \(-\frac{3}{2}\).
Question 2: 2. To simplify \( 8\sqrt{32} \), start by rewriting both expressions:
- \( 8 = 2^3 \)
- \( \sqrt{32} = \sqrt{2^5} = 2^{5/2} \)
Combining these gives: \[ 8\sqrt{32} = 2^3 \cdot 2^{5/2} = 2^{3 + 5/2} = 2^{\frac{6}{2} + \frac{5}{2}} = 2^{\frac{11}{2}} \] Thus, \( 8\sqrt{32} = 2^{\frac{11}{2}} \).
Question 3: 3. The expression \( 49\sqrt{7^{-3}} \) can be rewritten. \[ 49 = 7^2 \quad \text{and} \quad \sqrt{7^{-3}} = 7^{-3/2} \]
Therefore, \[ 49\sqrt{7^{-3}} = 7^2 \cdot 7^{-3/2} = 7^{2 + (-3/2)} = 7^{\frac{4}{2} - \frac{3}{2}} = 7^{\frac{1}{2}} \] The rational exponent for the new expression is \( \frac{1}{2} \).
Question 4: 4. Simplifying the left side of the equation: \[ 43 \cdot (64^{3/2}) = 43 \cdot (8^3) = 43 \cdot 512 = 43 \cdot 2^9 = 2^{4} \cdot 2^{9} = 2^{13}; \] The right side is: \[ 46 = 2^{6} \] Since \( 2^{13} \) does not equal \( 2^{6} \), the equation is False.
Question 5: 5. Simplifying the left side of the equation: \[ (25^{1/2})^4 = (5^2)^{4} = 5^{8}; \] The right side is: \[ 56 = 5^8 \] Since both sides are equal, the equation is True.
To summarize:
- The rational exponent is \(-\frac{3}{2}\).
- \(8\sqrt{32} = 2^{\frac{11}{2}}\).
- The rational exponent for the new expression is \(\frac{1}{2}\).
- The equation is False.
- The equation is True.