Problem 117 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{36}{35}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{6}{\sqrt{35}} \) or \( \frac{6\sqrt{35}}{35} \).

Step 1: Understand the expression

The given expression is \( \sqrt{\frac{36}{35}} \). This is a square root of a fraction.

Step 2: Apply the property of square roots to fractions

Use the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) to separate the square root of the numerator and the square root of the denominator. This gives us \( \sqrt{36} \) over \( \sqrt{35} \).

Step 3: Simplify the numerator

Calculate the square root of the numerator: \( \sqrt{36} = 6 \).

Step 4: Simplify the expression

Put the simplified numerator over the square root of the denominator: \( \frac{6}{\sqrt{35}} \).

Step 5: Rationalize the denominator (optional)

Multiply the numerator and the denominator by \( \sqrt{35} \) to rationalize the denominator: \( \frac{6 \sqrt{35}}{35} \).

Key Concepts

Square roots

Square roots are mathematical operations that seek to find a number which, when multiplied by itself, results in a given value. For example, the square root of 36 is 6 because 6 * 6 = 36. Square roots can be applied to both whole numbers and fractions. For a fraction, the square root of the numerator and the square root of the denominator can be separated, which is particularly useful when simplifying expressions.

Fractions

Fractions represent parts of a whole. A fraction consists of a numerator (top part) and a denominator (bottom part). In our example, we have the fraction \( \frac{36}{35} \). Using the rule of square roots for fractions, we get: \( \sqrt{\frac{36}{35}} = \frac{\sqrt{36}}{\sqrt{35}} \). This separates the fraction into two simpler parts where each part can be independently evaluated.

Rationalizing the denominator

Rationalizing the denominator means eliminating the square root from the bottom part of the fraction. This is done by multiplying both the numerator and the denominator by the same square root value found in the denominator. In our expression, \( \frac{6}{\sqrt{35}}\), we multiply both parts by \( \sqrt{35}\) to get \( \frac{6 \sqrt{35}}{35} \). The square root in the denominator is now gone, leaving us with a simpler, rationalized form of our expression.