Certainly! Let's go through each expression step-by-step and find the respective results.
1. \( 9^1 = 9 \)
- Any number raised to the power of 1 is the number itself. Therefore, \( 9^1 = 9 \).
2. \( 8^0 = 1 \)
- Any number raised to the power of 0 is 1. Therefore, \( 8^0 = 1 \).
3. \( x^1 = x \)
- Any variable raised to the power of 1 is the variable itself. Therefore, \( x^1 = x \).
4. \( x^0 = 1 \)
- Any variable raised to the power of 0 is 1. Therefore, \( x^0 = 1 \).
5. \(\left( \frac{3}{4} \right)^2 = 0.5625 \)
- Squaring \( \frac{3}{4} \) gives \( \left( \frac{3}{4} \right)^2 = \frac{9}{16} \), which equals \( 0.5625 \).
6. \( \frac{9^2}{3^2} = 9.0 \)
- Calculating separately, \( 9^2 = 81 \) and \( 3^2 = 9 \). Thus, \( \frac{81}{9} = 9.0 \).
7. \( \frac{5^2}{5^2} = 1.0 \)
- Since \( 5^2 = 25 \), the fraction simplifies to \( \frac{25}{25} = 1.0 \).
8. \( (7^3)(8^3) = 175616 \)
- First calculate \( 7^3 = 343 \) and \( 8^3 = 512 \). Then multiply: \( 343 \times 512 = 175616 \).
9. \( \sqrt{9^8} = 6561.0 \)
- Taking the square root of \( 9^8 \):
\( 9^8 = 43046721 \) and \( \sqrt{43046721} = 6561.0 \).
10. \( 4^3 \cdot 2^3 = 512 \)
- First calculate \( 4^3 = 64 \) and \( 2^3 = 8 \). Then multiply: \( 64 \times 8 = 512 \).
11. \( \left( 3^4 \right)^4 = 43046721 \)
- First calculate \( 3^4 = 81 \). Then raise it to the power of 4: \( 81^4 = 43046721 \).
12. \( (3^4)(3^5)(3^6)(3^2) = 129140163 \)
- Adding the exponents: \( 3^{4+5+6+2} = 3^{17} = 129140163 \).
13. \( (4^4)^4 = 4294967296 \)
- First calculate \( 4^4 = 256 \). Then raise it to the power of 4: \( 256^4 = 4294967296 \).
14. \( (5^3)(6^3)(10^3) = 27000000 \)
- First calculate separately \( 5^3 = 125 \), \( 6^3 = 216 \), and \( 10^3 = 1000 \). Then multiply: \( 125 \times 216 \times 1000 = 27000000 \).
15. \( \frac{50}{25} = 2.0 \)
- Dividing 50 by 25 gives \( \frac{50}{25} = 2.0 \).
In summary, we've evaluated each expression and obtained the results as follows:
[tex]\[
\begin{array}{ll}
9^1 = 9, & 8^0 = 1 \\
x^1 = x, & x^0 = 1 \\
\left( \frac{3}{4} \right)^2 = 0.5625, & \frac{9^2}{3^2} = 9.0 \\
\frac{5^2}{5^2} = 1.0, & (7^3)(8^3) = 175616 \\
\sqrt{9^8} = 6561.0, & 4^3 \cdot 2^3 = 512 \\
(3^4)^4 = 43046721, & (3^4)(3^5)(3^6)(3^2) = 129140163 \\
(4^4)^4 = 4294967296, & (5^3)(6^3)(10^3) = 27000000 \\
\frac{50}{25} = 2.0 &
\end{array}
\][/tex]
