Rationalizing Denominators
Master rationalizing denominators in Grade 10 algebra by multiplying by the conjugate to eliminate radicals from fractions and simplify expressions like 6/(4-√2).
Key Concepts
New Concept If $a$, $b$, $c$, and $d$ are rational numbers, then $a\sqrt{b} + c\sqrt{d}$ and $a\sqrt{b} c\sqrt{d}$ are conjugates of each other.
What’s next Next, you’ll use conjugates to rationalize binomial denominators, a key technique for simplifying complex radical expressions you'll see in future math and physics problems.
Common Questions
What is the conjugate and why is it used to rationalize?
The conjugate of a+b is a-b. Multiplying by the conjugate uses the difference of squares pattern (a+b)(a-b)=a²-b², which eliminates the radical from the denominator.
How do you rationalize the denominator of 6/(4-√2)?
Multiply numerator and denominator by the conjugate (4+√2): numerator becomes 24+6√2, denominator becomes 16-2=14, giving (12+3√2)/7.
What is the most common mistake when rationalizing denominators?
Multiplying only the denominator by the conjugate instead of both numerator and denominator. Always multiply top and bottom by the same expression to keep the fraction's value unchanged.