implify the \( \left(8 a^{3} b^{9}\right)^{1 / 3} \)

Certainly! Let's simplify the expression: \[ \left(8 a^{3} b^{9}\right)^{\frac{1}{3}} \] ### Step 1: Apply the Exponent to Each Factor When you have an expression raised to a power, you can distribute the exponent to each factor inside the parentheses: \[ \left(8 a^{3} b^{9}\right)^{\frac{1}{3}} = 8^{\frac{1}{3}} \times \left(a^{3}\right)^{\frac{1}{3}} \times \left(b^{9}\right)^{\frac{1}{3}} \] ### Step 2: Simplify Each Term Individually 1. **Simplify \(8^{\frac{1}{3}}\):** The cube root of 8 is 2 because \(2^3 = 8\). \[ 8^{\frac{1}{3}} = 2 \] 2. **Simplify \(\left(a^{3}\right)^{\frac{1}{3}}\):** When you raise a power to another power, you multiply the exponents. \[ \left(a^{3}\right)^{\frac{1}{3}} = a^{3 \times \frac{1}{3}} = a^{1} = a \] 3. **Simplify \(\left(b^{9}\right)^{\frac{1}{3}}\):** Similarly, multiply the exponents. \[ \left(b^{9}\right)^{\frac{1}{3}} = b^{9 \times \frac{1}{3}} = b^{3} \] ### Step 3: Combine the Simplified Terms Now, multiply the simplified results together: \[ 2 \times a \times b^{3} = 2 a b^{3} \] ### Final Simplified Form \[ \left(8 a^{3} b^{9}\right)^{\frac{1}{3}} = 2 a b^{3} \]