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!
