To solve the equation \( \sqrt[3]{4^{2x}} = 16 \), we will follow a step-by-step approach to isolate \( x \) and find its value. ### Step 1: Rewrite the Equation The equation given is: \[ \sqrt[3]{4^{2x}} = 16 \] We can rewrite the left side using the property of exponents. The cube root can be expressed as an exponent: \[ 4^{2x} = 16^3 \] ### Step 2: Simplify the Right Side Next, we need to simplify \( 16^3 \). We know that \( 16 \) can be expressed as a power of \( 4 \): \[ 16 = 4^2 \] Thus, we can rewrite \( 16^3 \) as: \[ 16^3 = (4^2)^3 = 4^{2 \cdot 3} = 4^6 \] ### Step 3: Set the Exponents Equal Now we have: \[ 4^{2x} = 4^6 \] Since the bases are the same (both are base \( 4 \)), we can set the exponents equal to each other: \[ 2x = 6 \] ### Step 4: Solve for \( x \) To find \( x \), we divide both sides of the equation by \( 2 \): \[ x = \frac{6}{2} = 3 \] ### Conclusion Thus, the value of \( x \) is \( 3 \). ### Verification To ensure our solution is correct, we can substitute \( x = 3 \) back into the original equation: \[ \sqrt[3]{4^{2 \cdot 3}} = \sqrt[3]{4^6} \] Calculating \( 4^6 \): \[ 4^6 = 4096 \] Now, taking the cube root: \[ \sqrt[3]{4096} = 16 \] This confirms that our solution is correct since both sides of the equation are equal. ### Explanation of Other Options - **Option A: 4** - If \( x = 4 \), then \( 2x = 8 \) and \( 4^{2x} = 4^8 \), which does not equal \( 16 \). - **Option B: 6** - If \( x = 6 \), then \( 2x = 12 \) and \( 4^{2x} = 4^{12} \), which is much larger than \( 16 \). - **Option D: 2** - If \( x = 2 \), then \( 2x = 4 \) and \( 4^{2x} = 4^4 = 256 \), which is also not equal to \( 16 \). ### Revision Summary - To solve equations involving roots and exponents, rewrite them in terms of the same base. - Set the exponents equal when the bases are the same. - Always verify your solution by substituting back into the original equation. - Be cautious with exponent rules and ensure calculations are accurate. The correct answer is **C. 3**.