Simplify: √45 + √20 − √5.

Correct — A. Pull the largest perfect square out of each root so that all three terms share √5.

√45 = √(9 × 5) = 3√5
√20 = √(4 × 5) = 2√5
√5 stays as √5

Combine like surds: 3√5 + 2√5 − √5 = (3 + 2 − 1)√5 = 4√5 → option (a)

Why the others are wrong

• (b)
  6√5 — 6√5 adds the last term instead of subtracting it: 3 + 2 + 1 = 6. The stem has − √5, so the coefficients give 3 + 2 − 1 = 4.
• (c)
  8√5 — 8√5 is twice the answer. The coefficients 3, 2 and −1 add to 4. As a decimal check, the expression is about 8.94, while 8√5 ≈ 17.9.
• (d)
  7√5 — 7√5 needs coefficients adding to 7. Here √45 gives 3, √20 gives 2, and subtracting √5 takes one away: 3 + 2 − 1 = 4.

Surds add only when the number under the root is the same, just like like terms in algebra: 3√5 + 2√5 = 5√5, but √45 + √20 is not √65.

To make roots alike, split each radicand into a perfect square × the rest, then use √(a²b) = a√b. Take the largest square that divides it.

Once every term is a multiple of the same root, add and subtract the coefficients only.

Key facts

• √(ab) = √a × √b for non-negative a and b.
• √a + √b is not √(a + b): √9 + √16 = 7, but √25 = 5.
• Only like surds, with the same number under the root, can be added or subtracted.

Common traps

• Adding the numbers under the roots: √45 + √20 − √5 is not √(45 + 20 − 5) = √60 ≈ 7.75.
• Dropping the minus sign on √5, which turns 4√5 into 6√5.

No directly related past PYQ was found.