Question

do not use the calculator for this question.

which expression is equivalent to \\(\sqrt{\frac{75}{16}}\\) in its simplest form?

a. \\(\frac{5}{4}\sqrt{3}\\)
b. \\(\sqrt{\frac{15 \cdot 5}{16}}\\)
c. \\(5\sqrt{\frac{3}{4}}\\)
d. \\(\sqrt{\frac{25\sqrt{3}}{16}}\\)

Explanation:

🆕 New Concept Discovered: Simplifying Radical Expressions
Pulling perfect squares out of square roots.

Step 1: Apply the quotient property of radicals

We can split the square root of a fraction into the square root of the numerator divided by the square root of the denominator:

$$ \sqrt{\frac{75}{16}} = \frac{\sqrt{75}}{\sqrt{16}} $$

Step 2: Simplify the denominator

The denominator contains a perfect square:

$$ \sqrt{16} = 4 $$

Substituting this back gives:

$$ \frac{\sqrt{75}}{4} $$

Step 3: Simplify the numerator

To simplify \(\sqrt{75}\), we look for the largest perfect square factor of \(75\). Since \(75 = 25 \times 3\) and \(25\) is a perfect square:

$$ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} $$

Step 4: Combine the simplified parts

Putting the simplified numerator and denominator together:

$$ \frac{5\sqrt{3}}{4} = \frac{5}{4}\sqrt{3} $$

This matches option A.

Answer:

A \(\frac{5}{4}\sqrt{3}\)