Remove radicals from denominators using conjugates. Math 20-1 Alberta.
Lesson 5.4 of Radical Expressions and Equations in Math 20-1 — Alberta curriculum lessons.
Convention 1 — No Negative in the Denominator. Fractions should not have a negative sign in the denominator. Instead, the negative goes in front of the fraction or in the numerator.
For example, (3)/(-4) is rewritten as -(3)/(4) by multiplying the top and bottom by -1:
(3)/(-4) = (3 × (-1))/((-4) × (-1)) = -(3)/(4)
Convention 2 — No Radical in the Denominator. Fractions should not have a radical sign in the denominator.
The same idea applies — multiply both the numerator and denominator by something carefully chosen so that the radical disappears from the bottom.
Definition — Rationalizing the Denominator. To rationalize a denominator, multiply both the numerator and the denominator by something that turns the denominator into a rational number — without changing the overall value of the fraction.
Multiplying both top and bottom by the same expression is the same as multiplying by 1, so the value stays the same.
Why Bother? Two Good Reasons. Reason 1 — Easier to compare. Rationalized forms are much easier to compare. For example, (1)/(√(2)) vs. (5)/(√(3)) is hard to evaluate mentally, but (√(2))/(2) vs. (5√(3))/(3) is much clearer.
Reason 2 — Historical convention. Before calculators existed, dividing by an irrational number was very difficult. Multiplying was far easier, so mathematicians always rationalized to avoid irrational denominators. The convention stuck.
Reminder — What Is a Conjugate?. Two binomials are conjugates if they have the same terms but the sign between them is different.
For example, (√(3) + 2) and (√(3) - 2) are conjugates. When you multiply conjugates, the cross terms cancel and the radicals disappear:
(√(3) + 2)(√(3) - 2) = (√(3))^2 - (2)^2 = 3 - 4 = -1
No radicals left — that is exactly what we want.
Example 1 — Rationalize (3)/(√(2)). Step 1: The denominator is √(2). Multiply top and bottom by √(2).
(3)/(√(2)) = (3 · √(2))/(√(2) · √(2))
Step 2: Simplify the denominator: √(2) · √(2) = 2.
= (3√(2))/(2)
Step 3: The numerator 3√(2) has no further simplification.
Answer: (3√(2))/(2)
Example 2 — Rationalize (√(5))/(√(6)). Step 1: The denominator is √(6). Multiply top and bottom by √(6).
(√(5))/(√(6)) = (√(5) · √(6))/(√(6) · √(6))
Step 2: Simplify top and bottom.
Top: √(5) · √(6) = √(30)
Bottom: √(6) · √(6) = 6
= (√(30))/(6)
Step 3: Check √(30): factors are 1, 2, 3, 5, 6, 10, 15, 30 — none (other than 1) are perfect squares. Already simplified.
Answer: (√(30))/(6)
Example 3 — Rationalize √((2)/(3)). Step 1: Split the radical using the quotient property √((a)/(b)) = (√(a))/(√(b)).
√((2)/(3)) = (√(2))/(√(3))
Step 2: Rationalize by multiplying top and bottom by √(3).
= (√(2) · √(3))/(√(3) · √(3))
Step 3: Simplify. Top: √(2) · √(3) = √(6). Bottom: √(3) · √(3) = 3.
Answer: (√(6))/(3)
Example 4 — Rationalize (6√(2))/(√(3)). Step 1: Multiply top and bottom by √(3).
(6√(2))/(√(3)) = (6√(2) · √(3))/(√(3) · √(3))
Step 2: Simplify. Top: 6√(2) · √(3) = 6√(6). Bottom: √(3) · √(3) = 3.
= (6√(6))/(3)
Step 3: Reduce by dividing 6 by 3.
Answer: 2√(6)
Example 5 — Rationalize (10√(2))/(√(8)). Step 1: Multiply top and bottom by √(8).
(10√(2))/(√(8)) = (10√(2) · √(8))/(√(8) · √(8))
Step 2: Simplify. Top: 10√(2) · √(8) = 10√(16) = 10 · 4 = 40. Bottom: √(8) · √(8) = 8.
= (40)/(8)
Step 3: Reduce.
Answer: 5
Verification: (10√(2))/(√(8)) = 10√((2)/(8)) = 10√((1)/(4)) = 10 · (1)/(2) = 5 ✓
Example 6 — Rationalize (2√(7))/(3√(2)). When the denominator has both a coefficient and a radical, you only need to multiply by the radical part — not the whole denominator.
Step 1: The radical in the denominator is √(2). Multiply top and bottom by √(2).
(2√(7))/(3√(2)) = (2√(7) · √(2))/(3√(2) · √(2))
Step 2: Simplify. Top: 2√(7) · √(2) = 2√(14). Bottom: 3√(2) · √(2) = 3 · 2 = 6.
= (2√(14))/(6)
Step 3: Reduce by dividing top and bottom by 2.
Answer: (√(14))/(3)
Example 7 — Rationalize (4√(8))/(2√(5)) (Simplify First). This problem benefits from simplifying before rationalizing.
Step 1: Reduce the coefficients: (4)/(2) = 2, so (4√(8))/(2√(5)) = (2√(8))/(√(5)).
Step 2: Multiply top and bottom by √(5).
= (2√(8) · √(5))/(√(5) · √(5)) = (2√(40))/(5)
Step 3: Simplify √(40). Largest perfect-square factor of 40 is 4:
√(40) = √(4 · 10) = 2√(10), so 2√(40) = 4√(10)
Answer: (4√(10))/(5)
Example 8 — Rationalize (8√(5))/(2√(10)) (Combine Radicals First). This one becomes much easier if we combine the radicals first.
Step 1: Reduce the coefficients: (8)/(2) = 4, so (8√(5))/(2√(10)) = (4√(5))/(√(10)).
Step 2: Combine the radicals using (√(a))/(√(b)) = √((a)/(b)).
= 4 · √((5)/(10)) = 4 · √((1)/(2))
Step 3: Split: √((1)/(2)) = (1)/(√(2)), so the expression becomes (4)/(√(2)).
Step 4: Rationalize by multiplying top and bottom by √(2).
= (4√(2))/(2)
Answer: 2√(2)
Example 9 — Rationalize (2√(5) + 4)/(√(10)) (Binomial Numerator). When the numerator is a binomial, rationalize the same way — the numerator just gets distributed.
Step 1: Multiply top and bottom by √(10).
((2√(5) + 4))/(√(10)) = ((2√(5) + 4) · √(10))/(√(10) · √(10))
Step 2: Distribute √(10) across the numerator.
Top: 2√(5) · √(10) + 4 · √(10) = 2√(50) + 4√(10)
Bottom: √(10) · √(10) = 10
Step 3: Simplify √(50) = √(25 · 2) = 5√(2), so 2√(50) = 10√(2).
= (10√(2) + 4√(10))/(10)
Step 4: Factor out 2 from the numerator: 10√(2) + 4√(10) = 2(5√(2) + 2√(10)).
= (2(5√(2) + 2√(10)))/(10)
Step 5: Reduce by dividing by 2.
Answer: (5√(2) + 2√(10))/(5)
Example 10 — Rationalize (2√(6) - 4√(3))/(√(3)). Step 1: Multiply top and bottom by √(3).
((2√(6) - 4√(3)))/(√(3)) = ((2√(6) - 4√(3)) · √(3))/(√(3) · √(3))
Step 2: Distribute √(3) across the numerator.
Top: 2√(6) · √(3) - 4√(3) · √(3) = 2√(18) - 4 · 3 = 2√(18) - 12
Bottom: √(3) · √(3) = 3
Step 3: Simplify √(18) = √(9 · 2) = 3√(2), so 2√(18) = 6√(2).
= (6√(2) - 12)/(3)
Step 4: Factor the numerator: 6√(2) - 12 = 6(√(2) - 2).
= (6(√(2) - 2))/(3)
Step 5: Reduce by dividing by 3: (6)/(3) = 2.
Answer: 2√(2) - 4
Tip — Simplify First When Possible. Looking for ways to reduce coefficients or simplify radicals before rationalizing usually leads to smaller numbers and less work. See Examples 7 and 8 for demonstrations of this strategy.
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