Let's tackle each of your questions step-by-step.
Question 3
You have the expression \( \frac{49}{7^{-3}} \).
By using the Quotient Property, which states that \( \frac{a^m}{a^n} = a^{m-n} \), we can rewrite \( \frac{49}{7^{-3}} \).
First, we know that \( 49 \) can be rewritten as \( 7^2 \): \[ \frac{49}{7^{-3}} = \frac{7^2}{7^{-3}} = 7^{2 - (-3)} = 7^{2 + 3} = 7^5 \]
Now, let's express this with a rational exponent. Recall that \( 7^5 \) is equivalent to \( 7^{5/1} \).
The rational exponent for the new expression is 5.
Question 4
We need to evaluate the equation \( 4^3 \cdot (64^{1/3})^2 = 4^6 \).
Start with the left side:
- \( 4^3 \): is simply \( 64 \).
- \( 64^{1/3} \) means taking the cube root of \( 64 \), which is \( 4 \). Therefore, \( (64^{1/3})^2 = 4^2 = 16 \).
Now, multiply \( 4^3 \) and \( 16 \): \[ 4^3 \cdot 16 = 64 \cdot 16 = 1024 = 4^5 \]
We can also check \( 4^6 \) on the right: \[ 4^6 = 4096 \]
Since \( 1024 \neq 4096 \), the equation \( 4^3 \cdot (64^{1/3})^2 = 4^6 \) is False.
Question 5
Now we evaluate the equation \( (25^{1/2})^4 \cdot 5^6 = 5^8 \).
Simplifying the left side:
- \( (25^{1/2})^4 \) means taking the square root of \( 25 \) and then raising it to the fourth power: \[ 25^{1/2} = 5 \quad \text{therefore,} \quad (25^{1/2})^4 = 5^4 \]
Now, multiply \( 5^4 \) by \( 5^6 \): \[ 5^4 \cdot 5^6 = 5^{4+6} = 5^{10} \]
Now, compare with the right side: \[ 5^8 \quad \text{is not equal to} \quad 5^{10} \]
So, the equation \( (25^{1/2})^4 \cdot 5^6 = 5^8 \) is also False.
Summary of Answers:
- Question 3: Rational exponent is 5.
- Question 4: The statement is False.
- Question 5: The statement is False.