$$ \begin{array}{l} 8^{-2}=\frac{1}{64}\ ext { h) }\ 0=\log _{6} 1\ \begin{array}{ll} \log _{x} x^{4}=4 \ m^{n}=p & \log _{6}\left(\frac{1}{36} ight)=-2 \ 3^{0}=1 & 2=\log _{1} x \ 10^{-3}=0,001 & \ \frac{1}{25}=5^{-2} & \ \left(\frac{1}{2} ight)^{-2}=4 & \end{array} \end{array} $$

Question

$$ \begin{array}{l} 8^{-2}=\frac{1}{64}\ 	ext { h) }\ 0=\log _{6} 1\ \begin{array}{ll} \log _{x} x^{4}=4 \ m^{n}=p & \log _{6}\left(\frac{1}{36}ight)=-2 \ 3^{0}=1 & 2=\log _{1} x \ 10^{-3}=0,001 & \ \frac{1}{25}=5^{-2} & \ \left(\frac{1}{2}ight)^{-2}=4 & \end{array} \end{array} $$

UpStudy AI · Accepted Answer

Let's analyze the given equations and statements step by step.

1. **First Statement:**
   $$
   8^{-2} = \frac{1}{64}
   $$
   This is correct. We can verify it:
   $$
   8^{-2} = \frac{1}{8^2} = \frac{1}{64}
   $$

2. **Second Statement:**
   $$
   0 = \log_{6} 1
   $$
   This is also correct. The logarithm of 1 in any base is 0:
   $$
   \log_{b} 1 = 0 \quad \text{for any } b > 0
   $$

3. **Third Statement:**
   $$
   \log_{x} x^{4} = 4
   $$
   This is true. By the properties of logarithms:
   $$
   \log_{b} b^{k} = k \quad \text{so here } k = 4
   $$

4. **Fourth Statement:**
   $$
   m^{n} = p
   $$
   This is a general expression and does not require verification without specific values.

5. **Fifth Statement:**
   $$
   \log_{6}\left(\frac{1}{36}\right) = -2
   $$
   We can verify this:
   $$
   \frac{1}{36} = 6^{-2} \implies \log_{6}(6^{-2}) = -2
   $$

6. **Sixth Statement:**
   $$
   3^{0} = 1
   $$
   This is correct. Any non-zero number raised to the power of 0 is 1.

7. **Seventh Statement:**
   $$
   2 = \log_{1} x
   $$
   This statement is incorrect. The logarithm base 1 is undefined because you cannot raise 1 to any power to get a number other than 1.

8. **Eighth Statement:**
   $$
   10^{-3} = 0.001
   $$
   This is correct. We can verify it:
   $$
   10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001
   $$

9. **Ninth Statement:**
   $$
   \frac{1}{25} = 5^{-2}
   $$
   This is correct. We can verify it:
   $$
   \frac{1}{25} = \frac{1}{5^2} = 5^{-2}
   $$

10. **Tenth Statement:**
    $$
    \left(\frac{1}{2}\right)^{-2} = 4
    $$
    This is correct. We can verify it:
    $$
    \left(\frac{1}{2}\right)^{-2} = \left(2\right)^{2} = 4
    $$

### Summary of Validity:
- Valid: 1, 2, 3, 5, 6, 8, 9, 10
- Invalid: 7

If you have any specific questions or need further assistance with any of these statements, feel free to ask!
The following statements are correct: 1. $$ 8^{-2} = \frac{1}{64} $$ 2. $$ 0 = \log_{6} 1 $$ 3. $$ \log_{x} x^{4} = 4 $$ 5. $$ \log_{6}\left(\frac{1}{36}\right) = -2 $$ 6. $$ 3^{0} = 1 $$ 8. $$ \frac{1}{25} = 5^{-2} $$ 10. $$ \left(\frac{1}{2}\right)^{-2} = 4 $$ The statement $$ 2 = \log_{1} x $$ is incorrect because the logarithm base 1 is undefined.

\( \begin{array}{l} 8^{-2}=\frac{1}{64}\\ \text { h) }\\ 0=\log _{6} 1\\ \begin{array}{ll} \log _{x} x^{4}=4 \\ m^{n}=p & \log _{6}\left(\frac{1}{36}\right)=-2 \\ 3^{0}=1 & 2=\log _{1} x \\ 10^{-3}=0,001 & \\ \frac{1}{25}=5^{-2} & \\ \left(\frac{1}{2}\right)^{-2}=4 & \end{array} \end{array} \)

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