A student simplifies 1/√5 by multiplying only the denominator by √5, getting 1/5. What error did the student make?
AThe student should have multiplied the denominator by 5, not √5
BThe student multiplied only the denominator by √5 without doing the same to the numerator, which changed the value of the expression
CThe student should have left the radical in the denominator since it is already in simplest form
DThe student used the wrong radical — the denominator should be multiplied by −√5
Rationalization works by multiplying both numerator and denominator by the same nonzero value — which is multiplying by 1, leaving the expression's value unchanged. Multiplying only the denominator divides by √5 without compensating the numerator, changing the value entirely. The correct rationalization: (1 · √5)/(√5 · √5) = √5/5. Verifying with a calculator that 1/√5 ≈ 0.447 and √5/5 ≈ 0.447 confirms the equivalence.
Question 2 Multiple Choice
To rationalize the denominator of 3/(2 + √7), what should you multiply numerator and denominator by?
A√7/√7, to eliminate the radical directly
B(2 − √7)/(2 − √7), the conjugate, to use the difference of squares pattern
C(2 + √7)/(2 + √7), the same expression, to square the denominator
D1/(2 − √7), to cancel the sum
When the denominator is a binomial containing a radical (a + √b), you multiply by the conjugate (a − √b)/(a − √b). The product (2 + √7)(2 − √7) = 4 − 7 = −3, which is rational — no radical remains. Multiplying by √7/√7 alone won't clear the 2 term. Squaring the original denominator creates a more complex irrational expression. The conjugate works precisely because it applies the difference of squares identity.
Question 3 True / False
Rationalizing the denominator changes the numerical value of the expression.
TTrue
FFalse
Answer: False
False. Rationalization multiplies the numerator and denominator by the same nonzero quantity — equivalent to multiplying the entire expression by 1. The numerical value is preserved exactly. You can verify: 1/√3 ≈ 0.577 and √3/3 ≈ 0.577. Same number, different written form. This is why the technique is valid: it transforms the expression's appearance without altering what it equals.
Question 4 True / False
To rationalize 1/(3 − √2), the correct approach is to multiply by √2/√2, since √2 is the main irrational part of the denominator.
TTrue
FFalse
Answer: False
False. When the denominator is a two-term expression (a − √b), the conjugate is (a + √b)/(a + √b) — the full two-term expression with the sign flipped. Multiplying (3 − √2)(3 + √2) = 9 − 2 = 7, eliminating the radical. Multiplying by √2/√2 alone gives (3√2 − 2)/2√2, which still has a radical in the denominator. The difference of squares identity requires the full conjugate, not just the radical part.
Question 5 Short Answer
Why is the conjugate technique guaranteed to produce a rational denominator when the denominator is of the form (a + √b)? What algebraic identity does it exploit?
Think about your answer, then reveal below.
Model answer: The conjugate technique exploits the difference of squares identity: (a + √b)(a − √b) = a² − (√b)² = a² − b. Since a and b are rational, a² − b is rational. The radical cancels because squaring √b eliminates the square root. The conjugate is specifically chosen to pair with the original expression and produce this square-difference, which has no remaining radical.
This is not a trick — it's a direct application of an identity students already know. The same pattern underlies many algebraic techniques: whenever you see (√a + √b), the conjugate (√a − √b) produces (√a)² − (√b)² = a − b, rational if a and b are rational. Recognizing this structure is what separates students who understand rationalization from those who only memorize the steps.