28

MULTIPLYING AND DIVIDING
RADICALS

Conjugate pairs

HERE IS THE RULE for multiplying radicals:

It is the symmetrical version of the rule for simplifying radicals. It is valid for a and b greater than or equal to 0.

Problem 1. Multiply.

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The difference of two squares

Problem 2. Multiply, then simplify:

multiply radicals

multiply radicals

Example 1. Multiply (multiply radicals + multiply radicals)(multiply radicals − multiply radicals).

Solution. The student should recognize the form those factors will produce:

The difference of two squares

Problem 3. Multiply.

a) (multiply radicals + multiply radicals)(multiply radicals − multiply radicals) = 5 − 3 = 2

b) (2multiply radicals + multiply radicals)(2multiply radicals − multiply radicals) = 4 · 3 − 6 = 12 − 6 = 6

c) (1 + multiply radicals)(1 − multiply radicals) = 1 − (x + 1) = 1 − x − 1 = −x

d) (multiply radicals + multiply radicals)(multiply radicals − multiply radicals) = a − b

Problem 4. (x − 1 − multiply radicals)(x − 1 + multiply radicals)

a) What form does that produce?

The difference of two squares. x − 1 is "a." multiply radicals is "b.">

b) Multiply out.

Problem 5. Multiply out.

Dividing radicals

For example,

Problem 6. Simplify the following.

Conjugate pairs

The conjugate of a + multiply radicals is a − multiply radicals. They are a conjugate pair.

Example 2. Multiply 6 − multiply radicals with its conjugate.

Solution. The product of a conjugate pair --

(6 − multiply radicals)(6 + multiply radicals)

-- is the difference of two squares. Therefore,

(6 − multiply radicals)(6 + multiply radicals) = 36 − 2 = 34.

When we multiply a conjugate pair, the radical vanishes and we obtain a rational number.

Problem 7. Multiply each number with its conjugate.

a) x + multiply radicals multiply radicals = x² − y

b) 2 − multiply radicals (2 − multiply radicals)(2 + multiply radicals) = 4 − 3 = 1

d) 4 − multiply radicals 16 − 5 = 11

Example 3. Rationalize the denominator:

Solution. Multiply both the denominator and the numerator by the conjugate of the denominator; that is, multiply them by 3 − multiply radicals.

The numerator becomes 3 − multiply radicals. The denominator becomes the difference of the two squares.

Problem 8. Write out the steps that show the following.

Problem 10. Here is a problem that comes up in calculus. Write out the steps that show:

In this case, you will have to rationalize the numerator.

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Next Lesson: Rational exponents

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