Question

Simplify the expression. $$\frac{1}{2} \sqrt{32} \cdot \sqrt{2}$$

Short answer

The simplified version of the expression \( \frac{1}{2} \sqrt{32} \cdot \sqrt{2} \) is 4.

Step-by-step solution

Simplify the Square Root of 32

First, simplify the square root of 32. This is done by looking for perfect square factors of 32. The largest perfect square factor of 32 is 16, so it is possible to write \( \sqrt{32} = \sqrt{16} \cdot \sqrt{2}\). Since the square root of 16 is 4, rewrite the expression as \( \frac{1}{2} \cdot 4 \sqrt{2}\).

Carry out the multiplication

Ensure your final expression does not contain any multiplication. Multiply \(\frac{1}{2}\) by 4 to simplify the expression to \(2 \sqrt{2}\).

Multiply the Simplified Expression by √2

The next step is to multiply our result from step 2, which is \(2 \sqrt{2}\), with \(\sqrt{2}\) from the original question. Making use of the product rule of radicals, which states that the multiplication of radicals results in the addition of the amounts under the radical, \((2 \sqrt{2}) \cdot \sqrt{2} = 2 \sqrt{4}\).

Simplify Final Result

Simplify the square root in \(2 \sqrt{4}\), since the square root of 4 is 2. The final result becomes \(2 \cdot 2 = 4\)