Rationalizing Denominators with Conjugates Solutions

• A    conjugate    is a pair of binomials with identical terms but opposite operators.
• Conjugates are written in the form: $\left(a\sqrt{b}–c\sqrt{d}\right) \mathrm{and} \left(a\sqrt{b}+c\sqrt{d}\right)$.
  • The terms are    identical    for both binomials.
  • But one pair    adds    and the other pair    subtracts    the terms.
• When the denominator contains a    binomial    expression with a    square    root, use the conjugate to rationalize the entire expression. 
• The product of the conjugates is always a    rational number ℚ   . 
• This statement is true because a    difference of two squares    pattern is formed where the middle term simplifies from the expression.

Note

Remember that if radicals need to be added or subtracted, they must have identical radicands in order to be combined.

Example 4

Name the expression that forms a conjugate pair. Multiply the expressions together to prove they are conjugates.

Note

Because conjugates multiplied together form the difference of two squares, you should be able to square the first and last terms using mental math.

Example 5

Simplify. Rationalize the denominator.

$\frac{z}{z+\sqrt{7}}$

Note

Remember that only identical binomials can simplify out of expressions. This is why $z^2$ is a term in the numerator and denominator.

Example 6

Simplify. Rationalize the denominator.

$\frac{5m}{2\sqrt{3}–8}$