\( \begin{array}{ll}\text { a) } 2^{m} \times 2^{n} \times 2^{-n} & \text { b) } a^{-m} \times a^{n} \times a^{m} \\ \text { c) }(a+b)^{-1} \times(a+b)^{-3} & \text { d) }(x+y) \times(x+y)^{-1} \\ \text { e) } \frac{12^{0}}{12^{0}} & \text { f) } \frac{12}{12^{0}} \\ \text { g) }\left(a^{2}\right)^{-1} \times 3 & \text { h) }\left(a^{-2}\right)^{-3} \\ \text { i) } \frac{a^{2}}{c} \times \frac{c^{2}}{a} & \text { j) }\left(\frac{a}{c}\right)^{-2} \times\left(\frac{c}{a}\right)^{2} \\ \text { k) } 3\left(10^{2}\right)^{-2} & \text { 1) } a\left(b^{a}\right)^{b}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(a^{-m}\times a^{n}\times a^{m}\) - step1: Multiply the terms: \(a^{-m+n+m}\) - step2: Add the terms: \(a^{n}\) Calculate or simplify the expression \( (x+y) \times (x+y)^{-1} \). Simplify the expression by following steps: - step0: Solution: \(\left(x+y\right)\left(x+y\right)^{-1}\) - step1: Express with a positive exponent: \(\left(x+y\right)\times \frac{1}{x+y}\) - step2: Reduce the fraction: \(1\times 1\) - step3: Multiply the terms: \(1\) Calculate or simplify the expression \( a\left(b^{a}\right)^{b} \). Simplify the expression by following steps: - step0: Solution: \(a\left(b^{a}\right)^{b}\) - step1: Multiply the exponents: \(ab^{ab}\) Calculate or simplify the expression \( \left(a^{-2}\right)^{-3} \). Simplify the expression by following steps: - step0: Solution: \(\left(a^{-2}\right)^{-3}\) - step1: Multiply the exponents: \(a^{-2\left(-3\right)}\) - step2: Multiply the numbers: \(a^{6}\) Calculate or simplify the expression \( \frac{12^{0}}{12^{0}} \). Calculate the value by following steps: - step0: Calculate: \(\frac{12^{0}}{12^{0}}\) - step1: Evaluate the power: \(\frac{1}{12^{0}}\) - step2: Evaluate the power: \(\frac{1}{1}\) - step3: Divide the terms: \(1\) Calculate or simplify the expression \( \frac{12}{12^{0}} \). Calculate the value by following steps: - step0: Calculate: \(\frac{12}{12^{0}}\) - step1: Evaluate the power: \(\frac{12}{1}\) - step2: Divide the terms: \(12\) Calculate or simplify the expression \( \left(a^{2}\right)^{-1} \times 3 \). Simplify the expression by following steps: - step0: Solution: \(\left(a^{2}\right)^{-1}\times 3\) - step1: Multiply the exponents: \(a^{2\left(-1\right)}\times 3\) - step2: Simplify: \(a^{-2}\times 3\) - step3: Reorder the terms: \(3a^{-2}\) - step4: Simplify: \(\frac{3}{a^{2}}\) Calculate or simplify the expression \( 2^{m} \times 2^{n} \times 2^{-n} \). Simplify the expression by following steps: - step0: Solution: \(2^{m}\times 2^{n}\times 2^{-n}\) - step1: Multiply the terms: \(2^{m+n-n}\) - step2: Calculate: \(2^{m}\) Calculate or simplify the expression \( (a+b)^{-1} \times (a+b)^{-3} \). Simplify the expression by following steps: - step0: Solution: \(\left(a+b\right)^{-1}\left(a+b\right)^{-3}\) - step1: Multiply the terms: \(\left(a+b\right)^{-4}\) - step2: Express with a positive exponent: \(\frac{1}{\left(a+b\right)^{4}}\) - step3: Expand the expression: \(\frac{1}{a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}}\) Calculate or simplify the expression \( \left(\frac{a}{c}\right)^{-2} \times \left(\frac{c}{a}\right)^{2} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{a}{c}\right)^{-2}\left(\frac{c}{a}\right)^{2}\) - step1: Evaluate the power: \(\frac{c^{2}}{a^{2}}\times \left(\frac{c}{a}\right)^{2}\) - step2: Rewrite the expression: \(\frac{c^{2}}{a^{2}}\times \frac{c^{2}}{a^{2}}\) - step3: Multiply the terms: \(\frac{c^{2}\times c^{2}}{a^{2}\times a^{2}}\) - step4: Multiply the terms: \(\frac{c^{4}}{a^{2}\times a^{2}}\) - step5: Multiply the terms: \(\frac{c^{4}}{a^{4}}\) Calculate or simplify the expression \( \frac{a^{2}}{c} \times \frac{c^{2}}{a} \). Simplify the expression by following steps: - step0: Solution: \(\frac{a^{2}}{c}\times \frac{c^{2}}{a}\) - step1: Reduce the fraction: \(ac\) Calculate or simplify the expression \( 3\left(10^{2}\right)^{-2} \). Calculate the value by following steps: - step0: Calculate: \(3\left(10^{2}\right)^{-2}\) - step1: Multiply the exponents: \(3\times 10^{2\left(-2\right)}\) - step2: Multiply the numbers: \(3\times 10^{-4}\) - step3: Evaluate the power: \(3\times \frac{1}{10000}\) - step4: Multiply: \(\frac{3}{10000}\) Here are the simplified results for each expression: a) \( 2^{m} \times 2^{n} \times 2^{-n} \) - **Step**: Combine the exponents. - **Result**: \( 2^{m} \) --- b) \( a^{-m} \times a^{n} \times a^{m} \) - **Step**: Combine the exponents. - **Result**: \( a^{n} \) --- c) \( (a+b)^{-1} \times (a+b)^{-3} \) - **Step**: Combine the exponents. - **Result**: \( (a+b)^{-4} = \frac{1}{(a+b)^{4}} \) --- d) \( (x+y) \times (x+y)^{-1} \) - **Step**: This is the product of a term and its reciprocal. - **Result**: \( 1 \) --- e) \( \frac{12^{0}}{12^{0}} \) - **Step**: Any non-zero number raised to the power of 0 is 1. - **Result**: \( 1 \) --- f) \( \frac{12}{12^{0}} \) - **Step**: Simplify using \( 12^{0} = 1 \). - **Result**: \( 12 \) --- g) \( \left(a^{2}\right)^{-1} \times 3 \) - **Step**: Apply the negative exponent. - **Result**: \( \frac{3}{a^{2}} \) --- h) \( \left(a^{-2}\right)^{-3} \) - **Step**: Apply the negative exponent. - **Result**: \( a^{6} \) --- i) \( \frac{a^{2}}{c} \times \frac{c^{2}}{a} \) - **Step**: Multiply the fractions. - **Result**: \( \frac{ac^{2}}{a^{2}} = ac \) --- j) \( \left(\frac{a}{c}\right)^{-2} \times \left(\frac{c}{a}\right)^{2} \) - **Step**: Apply the negative exponent and simplify. - **Result**: \( \frac{c^{4}}{a^{4}} \) --- k) \( 3\left(10^{2}\right)^{-2} \) - **Step**: Apply the negative exponent. - **Result**: \( \frac{3}{10000} \) or \( 0.0003 \) --- 1) \( a\left(b^{a}\right)^{b} \) - **Step**: Simplify the expression. - **Result**: \( ab^{ab} \) --- These results summarize the simplifications for each expression provided.