Question:** If \(m + n = 7\) and \(m^2 + n^2 = 29\), find \(m^3 + n^3\).

["# Finding (m^3 + n^3) Given (m + n = 7) and (m^2 + n^2 = 29)", "Understanding how to calculate (m^3 + n^3) from the values of (m + n) and (m^2 + n^2) is a valuable skill in algebra. In this article, we’ll solve the problem: If (m + n = 7) and (m^2 + n^2 = 29), find (m^3 + n^3), using efficient algebraic identities and step-by-step reasoning.", "---", "## Step 1: Recall the Key Identity", "To compute (m^3 + n^3), use the identity:", "[\nm^3 + n^3 = (m + n)^3 - 3mn(m + n)\n]", "This identity relies on expressing the sum of cubes in terms of the sum (m + n), the product (mn), and their powers.", "---", "## Step 2: Use Given Values", "We are given:", "[\nm + n = 7 \quad \ ext{and} \quad m^2 + n^2 = 29\n]", "We need (mn) to apply the identity. Use the square of sum identity:", "[\n(m + n)^2 = m^2 + 2mn + n^2\n]", "Substitute known values:", "[\n7^2 = 29 + 2mn\n]\n[\n49 = 29 + 2mn\n]", "---", "## Step 3: Solve for (mn)", "[\n2mn = 49 - 29 = 20\n]\n[\nmn = 10\n]", "---", "## Step 4: Plug into the Sum of Cubes Formula", "Now substitute into the identity:", "[\nm^3 + n^3 = (m + n)^3 - 3mn(m + n) = 7^3 - 3 \ imes 10 \ imes 7\n]\n[\n= 343 - 210 = 133\n]", "---", "## Conclusion", "Thus, the value of (m^3 + n^3) is:", "[\n\boxed{133}\n]", "---", "## Why This Method Works", "This approach efficiently combines algebraic identities and substitution to avoid directly solving for (m) and (n). It highlights how knowing the sum and sum of squares allows precise calculation of higher powers — a powerful technique in algebra and problem-solving.", "---", "Keywords: (m^3 + n^3) formula, algebra problem, (m + n = 7), (m^2 + n^2 = 29), sum of cubes identity, solve for (mn), efficient algebra, step-by-step derivation.\nMeta Description: Learn how to find (m^3 + n^3) given (m + n = 7) and (m^2 + n^2 = 29) using a key algebraic identity and solving for the missing product."]