Questions: Simplify and rewrite the square root of 48 times the square root of 125 divided by the square root of 250 times the square root of 2 in the form a square root of b, with whole numbers a and b, and a>1.

Solution Steps

To simplify the given expression \(\frac{\sqrt{48} \cdot \sqrt{125}}{\sqrt{250} \cdot \sqrt{2}}\) and rewrite it in the form \(a \sqrt{b}\), we can follow these steps:

1. Combine the radicals in the numerator and the denominator.
2. Simplify the resulting radicals.
3. Simplify the fraction by dividing the simplified radicals.

We start with the expression:

\[ \frac{\sqrt{48} \cdot \sqrt{125}}{\sqrt{250} \cdot \sqrt{2}} \]

Using the property of radicals, we can combine the radicals in the numerator and the denominator:

\[ \frac{\sqrt{48 \cdot 125}}{\sqrt{250 \cdot 2}} \]

Calculating the products inside the radicals:

• For the numerator:

\[ 48 \cdot 125 = 6000 \quad \Rightarrow \quad \sqrt{6000} \]

• For the denominator:

\[ 250 \cdot 2 = 500 \quad \Rightarrow \quad \sqrt{500} \]

Thus, we rewrite the expression as:

\[ \frac{\sqrt{6000}}{\sqrt{500}} \]

Now we simplify the fraction:

\[ \frac{\sqrt{6000}}{\sqrt{500}} = \sqrt{\frac{6000}{500}} = \sqrt{12} \]

Next, we can express \(\sqrt{12}\) in the form \(a \sqrt{b}\):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

Final Answer