Question

What is the product of the following expression in simplest radical form?

5√32 ⋅2√7

Answer 1

What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.

12√24 / 3√300

Answer 2

To simplify the expression \( 5\sqrt{32} \cdot 2\sqrt{7} \), we can start by rewriting the components we have.

First, multiply the constants and the square roots separately:

\[
5 \cdot 2 = 10
\]

Next, for the square roots:

\[
\sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7}
\]

Now, calculate \( 32 \cdot 7 \):

\[
32 \cdot 7 = 224
\]

So, now we have:

\[
5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224}
\]

Next, we simplify \( \sqrt{224} \). We start by factoring \( 224 \):

\[
224 = 16 \cdot 14 = 16 \cdot (2 \cdot 7)
\]

Since \( 16 \) is a perfect square (\( 4^2 \)), we can simplify this:

\[
\sqrt{224} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14}
\]

Now, substitute \( \sqrt{224} \) back into our expression:

\[
10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14}
\]

Thus, the product of the original expression in simplest radical form is:

\[
\boxed{40\sqrt{14}}
\]

Answer 3

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we can start by simplifying both the numerator and the denominator.

### Step 1: Simplifying the Numerator and Denominator

**Numerator:**

\[
12\sqrt{24}
\]

We can simplify \(\sqrt{24}\):

\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6}
\]

So,

\[
12\sqrt{24} = 12 \cdot 2\sqrt{6} = 24\sqrt{6}
\]

**Denominator:**

Now simplifying \(\sqrt{300}\):

\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3}
\]

Thus, the denominator becomes:

\[
3\sqrt{300} = 3 \cdot 10\sqrt{3} = 30\sqrt{3}
\]

### Step 2: Rewrite the Expression

Now, substituting back into the expression gives:

\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]

### Step 3: Simplifying the Quotient

Now we simplify \(\frac{24\sqrt{6}}{30\sqrt{3}}\):

- For the coefficients:

\[
\frac{24}{30} = \frac{4}{5}
\]

- For the square roots, we can rewrite it as:

\[
\frac{\sqrt{6}}{\sqrt{3}} = \sqrt{\frac{6}{3}} = \sqrt{2}
\]

### Step 4: Combine Results

Putting it all together, we have:

\[
\frac{24\sqrt{6}}{30\sqrt{3}} = \frac{4}{5} \cdot \sqrt{2} = \frac{4\sqrt{2}}{5}
\]

Since the denominator does not need rationalization (it is already a simple integer), the final answer in simplest radical form is:

\[
\boxed{\frac{4\sqrt{2}}{5}}
\]