1. Use the laws of exponents to simplify the following: a. \( \frac{a^{5} \div a^{3}}{a^{5}}= \) b. \( \frac{\left(f^{9}\right)^{2}}{f^{8}}= \) \[ \begin{array}{l} \frac{a^{5} \div a^{3}}{a^{5}}=\frac{a^{5}-3}{4^{5}}=\frac{a^{2}}{a^{5}} \\ =a \end{array} \] \( \square \) c. \( \left(x^{5}\right)^{3} \div x^{2}= \) d. \( \frac{b^{8}}{b^{2}} \times b^{3}= \) \( \square \) \( \square \) e. \( \frac{\left(e^{6}\right)^{2}}{e^{4}} \times e= \) f. \( \frac{h^{7} \div h^{3}}{\left(h^{2}\right)^{2}}= \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x^{5}\right)^{3}}{x^{2}}\) - step1: Multiply the exponents: \(\frac{x^{5\times 3}}{x^{2}}\) - step2: Multiply the numbers: \(\frac{x^{15}}{x^{2}}\) - step3: Reduce the fraction: \(\frac{x^{15-2}}{1}\) - step4: Simplify: \(x^{15-2}\) - step5: Divide the terms: \(x^{13}\) Calculate or simplify the expression \( (a^5 / a^3) / a^5 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\frac{a^{5}}{a^{3}}\right)}{a^{5}}\) - step1: Remove the parentheses: \(\frac{\frac{a^{5}}{a^{3}}}{a^{5}}\) - step2: Divide the terms: \(\frac{a^{2}}{a^{5}}\) - step3: Reduce the fraction: \(\frac{1}{a^{5-2}}\) - step4: Reduce the fraction: \(\frac{1}{a^{3}}\) Calculate or simplify the expression \( (h^7 / h^3) / (h^2)^2 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\frac{h^{7}}{h^{3}}\right)}{\left(h^{2}\right)^{2}}\) - step1: Remove the parentheses: \(\frac{\frac{h^{7}}{h^{3}}}{\left(h^{2}\right)^{2}}\) - step2: Multiply the exponents: \(\frac{\frac{h^{7}}{h^{3}}}{h^{2\times 2}}\) - step3: Multiply the numbers: \(\frac{\frac{h^{7}}{h^{3}}}{h^{4}}\) - step4: Divide the terms: \(\frac{h^{4}}{h^{4}}\) - step5: Divide the terms: \(1\) Calculate or simplify the expression \( (f^9)^2 / f^8 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(f^{9}\right)^{2}}{f^{8}}\) - step1: Multiply the exponents: \(\frac{f^{9\times 2}}{f^{8}}\) - step2: Multiply the numbers: \(\frac{f^{18}}{f^{8}}\) - step3: Reduce the fraction: \(\frac{f^{18-8}}{1}\) - step4: Simplify: \(f^{18-8}\) - step5: Divide the terms: \(f^{10}\) Calculate or simplify the expression \( b^8 / b^2 * b^3 \). Simplify the expression by following steps: - step0: Solution: \(\frac{b^{8}}{b^{2}}\times b^{3}\) - step1: Divide the terms: \(b^{6}\times b^{3}\) - step2: Simplify the expression: \(b^{6+3}\) - step3: Add the numbers: \(b^{9}\) Calculate or simplify the expression \( (e^6)^2 / e^4 * e \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(e^{6}\right)^{2}}{e^{4}}\times e\) - step1: Multiply the exponents: \(\frac{e^{6\times 2}}{e^{4}}\times e\) - step2: Multiply by \(a^{-n}:\) \(e^{6\times 2}\times e^{-4}\times e\) - step3: Multiply the numbers: \(e^{12}\times e^{-4}\times e\) - step4: Multiply the terms: \(e^{12-4+1}\) - step5: Calculate: \(e^{9}\) Let's simplify each expression step by step using the laws of exponents. ### a. \[ \frac{a^{5} \div a^{3}}{a^{5}} \] Using the law of exponents \( \frac{a^m}{a^n} = a^{m-n} \): \[ = \frac{a^{5-3}}{a^{5}} = \frac{a^{2}}{a^{5}} = a^{2-5} = a^{-3} = \frac{1}{a^{3}} \] ### b. \[ \frac{\left(f^{9}\right)^{2}}{f^{8}} \] Using the power of a power law \( (a^m)^n = a^{m \cdot n} \): \[ = \frac{f^{9 \cdot 2}}{f^{8}} = \frac{f^{18}}{f^{8}} = f^{18-8} = f^{10} \] ### c. \[ \left(x^{5}\right)^{3} \div x^{2} \] Using the power of a power law and the division law: \[ = \frac{x^{5 \cdot 3}}{x^{2}} = \frac{x^{15}}{x^{2}} = x^{15-2} = x^{13} \] ### d. \[ \frac{b^{8}}{b^{2}} \times b^{3} \] Using the division law and then multiplication: \[ = b^{8-2} \times b^{3} = b^{6} \times b^{3} = b^{6+3} = b^{9} \] ### e. \[ \frac{\left(e^{6}\right)^{2}}{e^{4}} \times e \] Using the power of a power law and then the division law: \[ = \frac{e^{6 \cdot 2}}{e^{4}} \times e = \frac{e^{12}}{e^{4}} \times e = e^{12-4} \times e = e^{8} \times e = e^{8+1} = e^{9} \] ### f. \[ \frac{h^{7} \div h^{3}}{\left(h^{2}\right)^{2}} \] Using the division law and the power of a power law: \[ = \frac{h^{7-3}}{h^{2 \cdot 2}} = \frac{h^{4}}{h^{4}} = h^{4-4} = h^{0} = 1 \] ### Summary of Results: - a. \( \frac{1}{a^{3}} \) - b. \( f^{10} \) - c. \( x^{13} \) - d. \( b^{9} \) - e. \( e^{9} \) - f. \( 1 \)