Questions: Solve the following equation for (x). [ 64^2 x=16 ] (x=) (square)

Solution Steps

To solve the equation \(64^{2x} = 16\), we can express both sides of the equation as powers of the same base. Notice that \(64\) is \(2^6\) and \(16\) is \(2^4\). By rewriting the equation in terms of base 2, we can equate the exponents and solve for \(x\).

The given equation is \(64^{2x} = 16\). We can express both 64 and 16 as powers of 2:

• \(64 = 2^6\)
• \(16 = 2^4\)

Thus, the equation becomes \((2^6)^{2x} = 2^4\).

Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the left side: \[ (2^6)^{2x} = 2^{12x} \] So the equation becomes: \[ 2^{12x} = 2^4 \]

Since the bases are the same, we can equate the exponents: \[ 12x = 4 \]

Divide both sides by 12 to solve for \(x\): \[ x = \frac{4}{12} = \frac{1}{3} \]

Final Answer