Simplifying radicals means rewriting roots in their cleanest equivalent form. It matters because simplified radicals are easier to compare, combine, and use in equations from geometry, algebra, physics, and engineering. A radical is usually considered simplified when no perfect-square factor remains under a square root, no radical is left in a denominator, and like radical terms have been combined.
Key Facts
- sqrt(ab) = sqrt(a)sqrt(b) for a >= 0 and b >= 0
- sqrt(a/b) = sqrt(a)/sqrt(b) for a >= 0 and b > 0
- sqrt(ka^2) = a sqrt(k) when a >= 0 and k has no perfect-square factor
- a sqrt(m) + b sqrt(m) = (a + b)sqrt(m)
- sqrt(50) = sqrt(25 · 2) = 5sqrt(2)
- 1/sqrt(a) = sqrt(a)/a for a > 0
Vocabulary
- Radical
- A radical is an expression that uses a root symbol, such as a square root or cube root.
- Radicand
- The radicand is the number or expression inside the radical sign.
- Perfect square
- A perfect square is a number that can be written as an integer multiplied by itself, such as 36 = 6 · 6.
- Like radicals
- Like radicals have the same index and the same radicand, such as 3sqrt(5) and 7sqrt(5).
- Rationalizing the denominator
- Rationalizing the denominator means rewriting a fraction so that no radical remains in the denominator.
Common Mistakes to Avoid
- Adding unlike radicals, such as sqrt(2) + sqrt(3) = sqrt(5), is wrong because radicals can only be combined when the radical parts match exactly.
- Forgetting to pull out the largest perfect-square factor makes answers incomplete because sqrt(72) should become 6sqrt(2), not just 3sqrt(8).
- Splitting a radical over addition, such as sqrt(a + b) = sqrt(a) + sqrt(b), is wrong because the product and quotient properties do not apply to sums.
- Leaving a radical in the denominator, such as 4/sqrt(5), is usually not considered simplified because it should be rationalized to 4sqrt(5)/5.
Practice Questions
- 1 Simplify sqrt(98) by factoring out the largest perfect square.
- 2 Simplify and combine: 3sqrt(12) + 2sqrt(27) - sqrt(48).
- 3 Explain why sqrt(18) + sqrt(8) can be combined after simplifying, but sqrt(18) + sqrt(7) cannot.