Multiplying Radicals Worksheet

Multiplying Radicals

Read each question carefully and show all your work. Simplify all radical expressions to their simplest form. Rationalize denominators where necessary.

1. Multiply and simplify the following radical expressions:

a) \( \sqrt{3} \cdot \sqrt{12} \)

b) \( (2\sqrt{5})(3\sqrt{10}) \)

c) \( \sqrt{6}(\sqrt{3} + \sqrt{2}) \)

2. Expand and simplify the following expressions:

a) \( (\sqrt{7} + 2)(\sqrt{7} - 2) \)

b) \( (3\sqrt{2} - 1)(2\sqrt{2} + 5) \)

3. Rationalize the denominator and simplify the expression:

a) \( \frac{1}{\sqrt{5}} \)

b) \( \frac{6}{3\sqrt{2}} \)

c) \( \frac{4}{2 + \sqrt{3}} \)

4. Which of the following is equivalent to \( \sqrt{18} \cdot \sqrt{2} \)?

5. Determine if each statement is True or False.

a) \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) is always true for non-negative real numbers a and b.

b) \( (\sqrt{x} + \sqrt{y})^2 = x + y \) for all non-negative real numbers x and y.

6. Complete the following statements.

a) To rationalize a binomial denominator with a radical, multiply the numerator and denominator by its  .

b) When multiplying radicals with the same index, we multiply the   together and the   together.