1.2 Simplify the following completely: \( 1.2 .1 \quad\left(4 a^{2}-2 a b+b^{2}\right)(2 a+b) \) \( 1.2 .2 \quad\left(8 x^{12}\right)^{-\frac{1}{3}} \times 2 x^{-4}-x^{-8} \) \( 1.2 .3 \quad \frac{7^{2 n+2}-3.7^{2 n+1}}{49^{n} .4} \)

Simplify the expression by following steps: - step0: Simplify the product: \(\left(4a^{2}-2ab+b^{2}\right)\left(2a+b\right)\) - step1: Simplify the product: \(8a^{3}+b^{3}\) Calculate or simplify the expression \( (8 x^{12})^{-\frac{1}{3}} \times 2 x^{-4}-x^{-8} \). Simplify the expression by following steps: - step0: Solution: \(\left(8x^{12}\right)^{-\frac{1}{3}}\times 2x^{-4}-x^{-8}\) - step1: Multiply the terms: \(\frac{1}{x^{8}}-x^{-8}\) - step2: Express with a positive exponent: \(\frac{1}{x^{8}}-\frac{1}{x^{8}}\) - step3: Transform the expression: \(\frac{1-1}{x^{8}}\) - step4: Subtract the numbers: \(\frac{0}{x^{8}}\) - step5: Calculate: \(0\) Calculate or simplify the expression \( \frac{7^{2 n+2}-3.7^{2 n+1}}{49^{n} .4} \). Simplify the expression by following steps: - step0: Solution: \(\frac{7^{2n+2}-3.7^{2n+1}}{49^{n}\times 0.4}\) - step1: Convert the expressions: \(\frac{7^{2n+2}-\left(\frac{37}{10}\right)^{2n+1}}{49^{n}\times 0.4}\) - step2: Multiply the terms: \(\frac{7^{2n+2}-\left(\frac{37}{10}\right)^{2n+1}}{0.4\times 49^{n}}\) - step3: Simplify the expression: \(\frac{5\times 7^{2n+2}-5\left(\frac{37}{10}\right)^{2n+1}}{2\times 49^{n}}\) - step4: Expand the expression: \(\frac{\frac{10^{2n+1}\times 7^{2n+2}-37^{2n+1}}{2\times 10^{2n}}}{2\times 49^{n}}\) - step5: Multiply by the reciprocal: \(\frac{10^{2n+1}\times 7^{2n+2}-37^{2n+1}}{2\times 10^{2n}}\times \frac{1}{2\times 49^{n}}\) - step6: Multiply the terms: \(\frac{10^{2n+1}\times 7^{2n+2}-37^{2n+1}}{2\times 10^{2n}\times 2\times 49^{n}}\) - step7: Multiply the terms: \(\frac{10^{2n+1}\times 7^{2n+2}-37^{2n+1}}{4\times 10^{2n}\times 49^{n}}\) Here are the simplified results for each part of the problem: ### 1.2.1 To simplify \( \left(4 a^{2}-2 a b+b^{2}\right)(2 a+b) \): 1. **Distribute** \( (2a + b) \) across \( (4a^2 - 2ab + b^2) \): - \( 4a^2 \cdot 2a = 8a^3 \) - \( 4a^2 \cdot b = 4a^2b \) - \( -2ab \cdot 2a = -4a^2b \) - \( -2ab \cdot b = -2ab^2 \) - \( b^2 \cdot 2a = 2ab^2 \) - \( b^2 \cdot b = b^3 \) 2. **Combine like terms**: - The \( 4a^2b \) and \( -4a^2b \) cancel out. - The remaining terms are \( 8a^3 + b^3 \). Thus, the simplified expression is: \[ 8a^{3}+b^{3} \] ### 1.2.2 To simplify \( (8 x^{12})^{-\frac{1}{3}} \times 2 x^{-4}-x^{-8} \): 1. **Calculate** \( (8 x^{12})^{-\frac{1}{3}} \): - \( 8^{-\frac{1}{3}} = 2^{-1} = \frac{1}{2} \) - \( (x^{12})^{-\frac{1}{3}} = x^{-4} \) Therefore, \( (8 x^{12})^{-\frac{1}{3}} = \frac{1}{2} x^{-4} \). 2. **Multiply** by \( 2 x^{-4} \): - \( \frac{1}{2} x^{-4} \times 2 x^{-4} = x^{-8} \). 3. **Combine** with \( -x^{-8} \): - \( x^{-8} - x^{-8} = 0 \). Thus, the simplified expression is: \[ 0 \] ### 1.2.3 To simplify \( \frac{7^{2 n+2}-3 \cdot 7^{2 n+1}}{49^{n} \cdot 4} \): 1. **Rewrite** \( 49^n \) as \( (7^2)^n = 7^{2n} \): - The denominator becomes \( 4 \cdot 7^{2n} \). 2. **Factor the numerator**: - \( 7^{2n+2} - 3 \cdot 7^{2n+1} = 7^{2n+1}(7 - 3) = 4 \cdot 7^{2n+1} \). 3. **Combine**: - The expression becomes \( \frac{4 \cdot 7^{2n+1}}{4 \cdot 7^{2n}} = \frac{7^{2n+1}}{7^{2n}} = 7^{1} = 7 \). Thus, the simplified expression is: \[ 7 \] ### Summary of Results: - **1.2.1**: \( 8a^{3}+b^{3} \) - **1.2.2**: \( 0 \) - **1.2.3**: \( 7 \)