Question

8 times radical 27 over 2 times radical 45

Answer 1

To simplify the expression \( \frac{8\sqrt{27}}{2\sqrt{45}} \), we can follow these steps:

1. Simplify the radicals in the expression.
2. Simplify the expression further.

### Step 1: Simplify the radicals

- First, simplify \( \sqrt{27} \):
\[
\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}
\]

- Next, simplify \( \sqrt{45} \):
\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\]

### Step 2: Substitute back into the expression
Now substitute these simplifications back into the original expression:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8 \cdot 3\sqrt{3}}{2 \cdot 3\sqrt{5}} = \frac{24\sqrt{3}}{6\sqrt{5}}
\]

### Step 3: Simplify the fraction
Next, simplify the fraction:
\[
\frac{24}{6} = 4
\]
So, we have:
\[
\frac{24\sqrt{3}}{6\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}}
\]

### Step 4: Rationalize the denominator (optional)
If needed, we can rationalize the denominator:
\[
4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{15}}{5} = \frac{4\sqrt{15}}{5}
\]

### Final Answer
Thus, the simplified form of the original expression is:
\[
\frac{4\sqrt{15}}{5}
\]