RATIONALIZE THE DENOMINATOR

Front

What does it mean to rationalize the denominator?

Back

To rationalize the denominator means to eliminate any radical (square root) expressions from the denominator of a fraction.

Front

Why is it important to rationalize the denominator?

Back

Rationalizing the denominator simplifies the expression and makes it easier to work with, especially in further calculations.

Front

What is the general form of a fraction that needs to be rationalized?

Back

A fraction of the form \( \frac{a}{\sqrt{b}} \) where \( b \) is a positive number.

Front

How do you rationalize the denominator of \( \frac{1}{\sqrt{2}} \)?

Back

Multiply the numerator and denominator by \( \sqrt{2} \): \( \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2} \).

Front

What is the result of rationalizing \( \frac{3}{\sqrt{5}} \)?

Back

Multiply by \( \frac{\sqrt{5}}{\sqrt{5}} \): \( \frac{3\sqrt{5}}{5} \).

Front

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

Back

For a binomial \( a + b \), multiply by the conjugate \( a - b \) to rationalize.

Front

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

Back

Multiply by \( \frac{3 - \sqrt{5}}{3 - \sqrt{5}} \): \( \frac{2(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{6 - 2\sqrt{5}}{4} = \frac{3 - \sqrt{5}}{2} \).

Front

What is the conjugate of \( a + b \)?

Back

The conjugate is \( a - b \).

Front

How do you simplify \( \frac{2x^3y\sqrt{30}}{y^4\\sqrt{3x}} \)?

Back

Factor out common terms and rationalize: \( \frac{2x^3\sqrt{30}}{y^3\sqrt{3x}} = \frac{2x^2\sqrt{10x}}{y^3} \).

Front

What is the first step in rationalizing a denominator?

Back

Identify if the denominator contains a radical and determine if it can be simplified.