IDNLearn.com is designed to help you find reliable answers quickly and easily. Our platform provides prompt, accurate answers from experts ready to assist you with any question you may have.

\begin{tabular}{|l|l|l|}
\hline
1. [tex]$2^{-4}$[/tex] & 2. [tex]$4^{-2}$[/tex] & 3. [tex]$x^{-6}$[/tex] \\
\hline
4. [tex]$3 z^{-2}$[/tex] & 5. [tex]$\frac{1}{3^{-2}}$[/tex] & 6. [tex]$5^0$[/tex] \\
\hline
7. [tex]$2^{-5} \cdot 2^3$[/tex] & 8. [tex]$x^3 \cdot x^{-7}$[/tex] & 9. [tex]$\frac{3^3}{3^5}$[/tex] \\
\hline
10. [tex]$\frac{x^4}{x^{-6}}$[/tex] & 11. [tex]$x^0$[/tex] & 12. [tex]$1001^{-1}$[/tex] \\
\hline
\end{tabular}


Sagot :

Let's solve each of the expressions in the table step-by-step:

1. [tex]\( 2^{-4} \)[/tex]
- Recall that a negative exponent means reciprocal, so [tex]\( 2^{-4} = \frac{1}{2^4} \)[/tex].
- Calculating [tex]\( 2^4 = 16 \)[/tex], we get [tex]\( \frac{1}{16} \)[/tex].
- Therefore, [tex]\( 2^{-4} = 0.0625 \)[/tex].

2. [tex]\( 4^{-2} \)[/tex]
- Similarly, [tex]\( 4^{-2} = \frac{1}{4^2} \)[/tex].
- Calculating [tex]\( 4^2 = 16 \)[/tex], we get [tex]\( \frac{1}{16} \)[/tex].
- Therefore, [tex]\( 4^{-2} = 0.0625 \)[/tex].

3. [tex]\( x^{-6} \)[/tex]
- An expression with a variable remains in its general form.
- So, [tex]\( x^{-6} \)[/tex] remains [tex]\( x^{-6} \)[/tex].

4. [tex]\( 3z^{-2} \)[/tex]
- This is interpreted as [tex]\( 3 \cdot z^{-2} \)[/tex], which is [tex]\( \frac{3}{z^2} \)[/tex].
- In its simplified form, it is [tex]\( 3 \cdot z^{-2} \)[/tex].

5. [tex]\( \frac{1}{3^{-2}} \)[/tex]
- Recall that [tex]\( 3^{-2} = \frac{1}{3^2} \)[/tex], so [tex]\( \frac{1}{3^{-2}} = 3^2 \)[/tex].
- Calculating [tex]\( 3^2 = 9 \)[/tex], we get [tex]\( \frac{1}{3^{-2}} = 9.0 \)[/tex].

6. [tex]\( 5^0 \)[/tex]
- Any non-zero number raised to the power of 0 is 1.
- Therefore, [tex]\( 5^0 = 1 \)[/tex].

7. [tex]\( 2^{-5} \cdot 2^3 \)[/tex]
- Using properties of exponents: [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
- So, [tex]\( 2^{-5} \cdot 2^3 = 2^{-5+3} = 2^{-2} \)[/tex].
- Calculating [tex]\( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)[/tex].
- Therefore, [tex]\( 2^{-5} \cdot 2^3 = 0.25 \)[/tex].

8. [tex]\( x^3 \cdot x^{-7} \)[/tex]
- Using properties of exponents: [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
- So, [tex]\( x^3 \cdot x^{-7} = x^{3-7} = x^{-4} \)[/tex].
- Therefore, [tex]\( x^3 \cdot x^{-7} = x^{-4} \)[/tex].

9. [tex]\( \frac{3^3}{3^5} \)[/tex]
- Using properties of exponents: [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex].
- So, [tex]\( \frac{3^3}{3^5} = 3^{3-5} = 3^{-2} \)[/tex].
- Calculating [tex]\( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)[/tex].
- Therefore, [tex]\( \frac{3^3}{3^5} = 0.1111111111111111 \)[/tex].

10. [tex]\( \frac{x^4}{x^{-6}} \)[/tex]
- Using properties of exponents: [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex].
- So, [tex]\( \frac{x^4}{x^{-6}} = x^{4-(-6)} = x^{4+6} = x^{10} \)[/tex].
- Therefore, [tex]\( \frac{x^4}{x^{-6}} = x^{10} \)[/tex].

11. [tex]\( x^0 \)[/tex]
- Any non-zero number raised to the power of 0 is 1.
- Therefore, [tex]\( x^0 = 1 \)[/tex].

12. [tex]\( 1001^{-1} \)[/tex]
- Recall that a negative exponent means reciprocal, so [tex]\( 1001^{-1} = \frac{1}{1001} \)[/tex].
- In its fraction form, it is approximately equal to 0.000999000999000999.
- Therefore, [tex]\( 1001^{-1} = 0.000999000999000999 \)[/tex].

If you have any questions or need further clarification, feel free to ask!