Question

simplify the radical.
$4\sqrt{80} = \square$

Explanation:

Step1: Factor 80 into perfect square and other

We know that \( 80 = 16\times5 \), where 16 is a perfect square (\( 16 = 4^2 \)). So we can rewrite \( \sqrt{80} \) as \( \sqrt{16\times5} \).

Step2: Use property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we have \( \sqrt{16\times5}=\sqrt{16}\times\sqrt{5} \). Since \( \sqrt{16} = 4 \), this becomes \( 4\sqrt{5} \).

Step3: Multiply by the coefficient 4

Now we have \( 4\times\sqrt{80}=4\times4\sqrt{5} \). Multiplying 4 and 4 gives 16, so the simplified form is \( 16\sqrt{5} \).

Answer:

\( 16\sqrt{5} \)