Questions: (4 x^3)^5 ⋅ (2 x^2)^-4

Solution Steps

To simplify the expression \((4x^3)^5 \cdot (2x^2)^{-4}\), we can use the properties of exponents. First, apply the power of a power rule to each term, which states that \((a^m)^n = a^{m \cdot n}\). Then, multiply the resulting expressions together by adding the exponents of like bases.

To simplify \((4x^3)^5\), apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\). This gives: \[ (4x^3)^5 = 4^5 \cdot x^{3 \cdot 5} = 1024 \cdot x^{15} \]

Similarly, for \((2x^2)^{-4}\): \[ (2x^2)^{-4} = 2^{-4} \cdot x^{-8} = \frac{1}{16} \cdot x^{-8} \]

Now, multiply the results from Step 1: \[ 1024 \cdot x^{15} \cdot \frac{1}{16} \cdot x^{-8} = \left(1024 \cdot \frac{1}{16}\right) \cdot x^{15 + (-8)} \]

Calculate the numerical coefficient: \[ 1024 \cdot \frac{1}{16} = 64 \]

Combine the exponents of \(x\): \[ x^{15 + (-8)} = x^7 \]

Final Answer