Math 10 Rationalize the Denominator

Front

What does it mean to rationalize the denominator?

Back

Rationalizing the denominator means to eliminate any irrational numbers (like square roots) from the denominator of a fraction.

Front

How do you rationalize a denominator that contains a square root?

Back

Multiply the numerator and denominator by the square root present in the denominator.

Front

What is the result of rationalizing \( \frac{1}{\sqrt{2}} \)?

Back

\( \frac{1}{\sqrt{2}} \) becomes \( \frac{\sqrt{2}}{2} \) after rationalization.

Front

What is the formula for rationalizing a denominator with a binomial?

Back

For a binomial like \( a + b \), multiply by the conjugate \( a - b \) in both the numerator and denominator.

Front

Rationalize \( \frac{3}{2 + \sqrt{5}} \).

Back

Multiply by the conjugate: \( \frac{3(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} = \frac{3(2 - \sqrt{5})}{4 - 5} = -3(2 - \sqrt{5}) \).

Front

What is the conjugate of \( 3 + \sqrt{7} \)?

Back

The conjugate is \( 3 - \sqrt{7} \).

Front

Why is it important to rationalize the denominator?

Back

Rationalizing the denominator can simplify calculations and make the expression easier to understand.

Front

Rationalize \( \frac{5}{\sqrt{3} + 1} \).

Back

Multiply by the conjugate: \( \frac{5(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{5(\sqrt{3} - 1)}{3 - 1} = \frac{5(\sqrt{3} - 1)}{2} \).

Front

What happens when you multiply a fraction by 1?

Back

The value of the fraction remains unchanged, as multiplying by 1 does not alter its value.

Front

Rationalize \( \frac{4}{\sqrt{6}} \).

Back

Multiply by \( \frac{\sqrt{6}}{\sqrt{6}} \): \( \frac{4\sqrt{6}}{6} = \frac{2\sqrt{6}}{3} \).