1. **Problem:** Calculate $2^3$. **Solution:** Use the power rule: $a^n$ means multiply $a$ by itself $n$ times. $2^3 = 2 \times 2 \times 2 = 8$. 2. **Problem:** Simplify $5^0$. **Solution:** Any non-zero number raised to the power 0 is 1. $5^0 = 1$. 3. **Problem:** Find the value of $3^2 \times 3^3$. **Solution:** When multiplying powers with the same base, add exponents: $3^2 \times 3^3 = 3^{2+3} = 3^5 = 243$. 4. **Problem:** Simplify $\frac{7^5}{7^2}$. **Solution:** When dividing powers with the same base, subtract exponents: $\frac{7^5}{7^2} = 7^{5-2} = 7^3 = 343$. 5. **Problem:** Calculate $(2^3)^4$. **Solution:** When raising a power to another power, multiply exponents: $(2^3)^4 = 2^{3 \times 4} = 2^{12} = 4096$. 6. **Problem:** Simplify $10^{-2}$. **Solution:** Negative exponent means reciprocal: $10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01$. 7. **Problem:** Find $4^3 \times 2^3$. **Solution:** Since bases differ but exponents are same, multiply bases then raise to power: $4^3 \times 2^3 = (4 \times 2)^3 = 8^3 = 512$. 8. **Problem:** Simplify $\left(\frac{3}{4}\right)^2$. **Solution:** Raise numerator and denominator to power 2: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$. 9. **Problem:** Calculate $9^{\frac{1}{2}}$. **Solution:** Fractional exponent $\frac{1}{2}$ means square root: $9^{\frac{1}{2}} = \sqrt{9} = 3$. 10. **Problem:** Simplify $16^{\frac{3}{4}}$. **Solution:** Write as $(16^{\frac{1}{4}})^3$, $16^{\frac{1}{4}}$ is fourth root of 16: $16^{\frac{1}{4}} = 2$, so $16^{\frac{3}{4}} = 2^3 = 8$. 11. **Problem:** Find $2^4 \div 2^2$. **Solution:** Subtract exponents: $2^4 \div 2^2 = 2^{4-2} = 2^2 = 4$. 12. **Problem:** Simplify $\left(5^2 \times 5^3\right)^2$. **Solution:** Multiply inside first: $5^2 \times 5^3 = 5^{2+3} = 5^5$, then raise to power 2: $(5^5)^2 = 5^{5 \times 2} = 5^{10} = 9765625$. 13. **Problem:** Calculate $\left(\frac{1}{2}\right)^{-3}$. **Solution:** Negative exponent means reciprocal: $\left(\frac{1}{2}\right)^{-3} = \left(2\right)^3 = 8$. 14. **Problem:** Simplify $8^{\frac{2}{3}}$. **Solution:** Write as $(8^{\frac{1}{3}})^2$, cube root of 8 is 2: $8^{\frac{2}{3}} = 2^2 = 4$. 15. **Problem:** Find the value of $\frac{2^5 \times 3^5}{6^5}$. **Solution:** Write denominator as $(2 \times 3)^5 = 2^5 \times 3^5$, so $\frac{2^5 \times 3^5}{6^5} = \frac{2^5 \times 3^5}{2^5 \times 3^5} = 1$. 16. **Problem:** Simplify $\left(\frac{4}{9}\right)^{-\frac{1}{2}}$. **Solution:** Negative exponent means reciprocal and $\frac{1}{2}$ means square root: $\left(\frac{4}{9}\right)^{-\frac{1}{2}} = \left(\frac{9}{4}\right)^{\frac{1}{2}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$. 17. **Problem:** Calculate $27^{\frac{2}{3}}$. **Solution:** Cube root of 27 is 3, then square: $27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2 = 3^2 = 9$. 18. **Problem:** Simplify $\frac{(2^3)^2}{2^4}$. **Solution:** $(2^3)^2 = 2^{3 \times 2} = 2^6$, so $\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$. 19. **Problem:** Find $5^{-1} + 5^0$. **Solution:** $5^{-1} = \frac{1}{5} = 0.2$, $5^0 = 1$, so sum is $0.2 + 1 = 1.2$. 20. **Problem:** Simplify $\left(3^2 \times 4^2\right)^{\frac{1}{2}}$. **Solution:** Multiply inside: $3^2 \times 4^2 = 9 \times 16 = 144$, then square root: $\sqrt{144} = 12$.