Question

which expression is equivalent to the given expression? $2sqrt{20} cdot 4sqrt{6}$
a. $16sqrt{30}$
b. $sqrt{120}$
c. $32sqrt{30}$
d. $3sqrt{20}$
reset selection

Explanation:

Step1: Multiply the coefficients

First, multiply the coefficients of the square roots, which are 2 and 4. So, \(2\times4 = 8\)? Wait, no, wait. Wait, the original expression is \(2\sqrt{20}\cdot4\sqrt{6}\). Wait, no, maybe I misread. Wait, let's re-express. The expression is \(2\sqrt{20} \times 4\sqrt{6}\). So, multiply the coefficients: \(2\times4 = 8\)? Wait, no, wait, maybe the problem is \(2\sqrt{20} \cdot 4\sqrt{6}\), but let's check the options. Wait, maybe I made a mistake. Wait, let's compute \(2\sqrt{20} \times 4\sqrt{6}\).

First, multiply the coefficients: \(2 \times 4 = 8\). Then, multiply the square roots: \(\sqrt{20} \times \sqrt{6} = \sqrt{20 \times 6} = \sqrt{120}\). So now we have \(8\sqrt{120}\)? Wait, no, that's not matching the options. Wait, maybe I misread the original expression. Wait, the options have 16√30, 32√30, etc. Wait, maybe the original expression is \(2\sqrt{20} \times 4\sqrt{6}\), but let's factor 20 and 6.

Wait, 20 can be factored into 4×5, so \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\). Then, the expression becomes \(2 \times 2\sqrt{5} \times 4\sqrt{6}\). Wait, no, that's not right. Wait, let's do it step by step.

Original expression: \(2\sqrt{20} \times 4\sqrt{6}\)

Multiply the coefficients: \(2 \times 4 = 8\)

Multiply the radicands: \(\sqrt{20} \times \sqrt{6} = \sqrt{20 \times 6} = \sqrt{120}\)

So now we have \(8\sqrt{120}\). But 120 can be factored into 4×30, so \(\sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30}\). Then, substitute back: \(8 \times 2\sqrt{30} = 16\sqrt{30}\). Ah, there we go. So let's verify:

1. Multiply coefficients: \(2 \times 4 = 8\)
2. Multiply radicands: \(\sqrt{20} \times \sqrt{6} = \sqrt{120}\)
3. Simplify \(\sqrt{120}\): \(\sqrt{4 \times 30} = 2\sqrt{30}\)
4. Multiply 8 and 2: \(8 \times 2 = 16\)
5. So the result is \(16\sqrt{30}\)

Step2: Check the options

Looking at the options, option A is \(16\sqrt{30}\), which matches our result.

Answer:

A. \(16\sqrt{30}\)