Question 3a Find the first three terms, in ascending powers of \( X \), in the expansion of \( (3+x)^{4} \). Question 3b Use your answer to part (a) to estimate \( (3.1)^{4} \).

To find the first three terms in ascending powers of \( x \) in the expansion of \( (3+x)^{4} \), we can use the binomial theorem. The binomial theorem states that for any positive integer \( n \) and any real numbers \( a \) and \( b \), the expansion of \( (a+b)^{n} \) is given by: \[ (a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \] where \( \binom{n}{k} \) represents the binomial coefficient. For \( (3+x)^{4} \), we have \( a = 3 \) and \( b = x \), and \( n = 4 \). The first three terms in ascending powers of \( x \) will be the terms with \( k = 0, 1, 2 \). Let's calculate the first three terms using the binomial theorem: 1. For \( k = 0 \): \[ \binom{4}{0} 3^{4-0} x^{0} = 1 \cdot 3^{4} \cdot 1 = 81 \] 2. For \( k = 1 \): \[ \binom{4}{1} 3^{4-1} x^{1} = 4 \cdot 3^{3} \cdot x = 108x \] 3. For \( k = 2 \): \[ \binom{4}{2} 3^{4-2} x^{2} = 6 \cdot 3^{2} \cdot x^{2} = 54x^{2} \] Therefore, the first three terms in ascending powers of \( x \) in the expansion of \( (3+x)^{4} \) are: \[ 81 + 108x + 54x^{2} \] Now, to estimate \( (3.1)^{4} \) using the first three terms we found, we can substitute \( x = 0.1 \) into the expression \( 81 + 108x + 54x^{2} \) and calculate the result. Calculate the value by following steps: - step0: Calculate: \(81+108\times 0.1+54\times 0.1^{2}\) - step1: Convert the expressions: \(81+108\times 0.1+54\left(\frac{1}{10}\right)^{2}\) - step2: Multiply the numbers: \(81+10.8+54\left(\frac{1}{10}\right)^{2}\) - step3: Multiply the terms: \(81+10.8+\frac{27}{50}\) - step4: Add the numbers: \(91.8+\frac{27}{50}\) - step5: Convert the expressions: \(\frac{459}{5}+\frac{27}{50}\) - step6: Reduce fractions to a common denominator: \(\frac{459\times 10}{5\times 10}+\frac{27}{50}\) - step7: Calculate: \(\frac{4590}{50}+\frac{27}{50}\) - step8: Transform the expression: \(\frac{4590+27}{50}\) - step9: Add the numbers: \(\frac{4617}{50}\) The estimated value of \( (3.1)^{4} \) using the first three terms in the expansion of \( (3+x)^{4} \) is approximately 92.34.