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Simplify each of the expressions below. \( \begin{array}{ll}\text { 1. }\left(5^{3}\right)^{3} 39 & 4 . \\ \left(3^{4}\right)^{2}\end{array} \)

Ask by Haynes Long. in the United States
Feb 24,2025

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\( (5^3)^3 = 5^9 = 1,953,125 \) \( (3^4)^2 = 3^8 = 6,561 \)

Solución

We are given two expressions: 1. (5³)³ 2. (3⁴)² To simplify these expressions, use the exponent rule for powers raised to powers:   (a^b)^c = a^(b × c) Step 1: Simplify each expression using the rule. For the first expression:   (5³)³ = 5^(3 × 3) = 5⁹ For the second expression:   (3⁴)² = 3^(4 × 2) = 3⁸ Step 2: (Optional) Compute the numerical values. For 5⁹: 5⁹ = 5 × 5 × ... (9 times) Calculating, 5⁹ = 1,953,125 For 3⁸: 3⁸ = 3 × 3 × ... (8 times) Calculating, 3⁸ = 6,561 Thus, the simplified forms are: 1. (5³)³ = 5⁹ = 1,953,125 2. (3⁴)² = 3⁸ = 6,561

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Beyond the Answer

To simplify the expressions, we can use the power of a power property, which states that \((a^m)^n = a^{m \cdot n}\). 1. For \((5^{3})^{3}\), we apply the rule: \[ (5^{3})^{3} = 5^{3 \cdot 3} = 5^{9} \] 2. For \((3^{4})^{2}\), we also use the same rule: \[ (3^{4})^{2} = 3^{4 \cdot 2} = 3^{8} \] So, the simplified expressions are \(5^9\) and \(3^8\).

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