Question

What is the quotient of the fallowing expression? State the answer in simplest form. Rationalize the denominator if necessary.
9* the square root of 80 devided by 4 times the squaroot of 90

Answer 1

To simplify the expression \(\frac{9 \sqrt{80}}{4 \sqrt{90}}\), let's begin by simplifying both the numerator and the denominator.

**Step 1: Simplify the square roots.**

1. **Simplifying \(\sqrt{80}\):**
\[
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5}
\]

2. **Simplifying \(\sqrt{90}\):**
\[
\sqrt{90} = \sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3 \sqrt{10}
\]

**Step 2: Substitute the simplified square roots back into the expression.**
Now, substitute back into the original expression:
\[
\frac{9 \sqrt{80}}{4 \sqrt{90}} = \frac{9 \cdot 4 \sqrt{5}}{4 \cdot 3 \sqrt{10}}
\]

**Step 3: Cancel the \(4\) in the numerator and denominator:**
\[
= \frac{9 \sqrt{5}}{3 \sqrt{10}}
\]

**Step 4: Simplify the fraction:**
Now, divide both numerator and denominator by \(3\):
\[
= \frac{3 \sqrt{5}}{\sqrt{10}}
\]

**Step 5: Rationalize the denominator:**
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{10}\):
\[
= \frac{3 \sqrt{5} \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} = \frac{3 \sqrt{50}}{10}
\]

**Step 6: Simplify \(\sqrt{50}\):**
\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2}
\]
So substitute back:
\[
= \frac{3 \cdot 5 \sqrt{2}}{10} = \frac{15 \sqrt{2}}{10}
\]

**Step 7: Simplify the fraction:**
Dividing both the numerator and the denominator by \(5\):
\[
= \frac{3 \sqrt{2}}{2}
\]

Thus, the final answer in simplest form is:
\[
\frac{3 \sqrt{2}}{2}
\]