5.2 Radicals, Rational Exponents & Complex Numbers

Simplifying Radicals

A radical like √x asks for a number whose square is x. To simplify a square root, factor the radicand so one factor is the largest perfect square, then take its root out front.

Worked example. Simplify √72. The largest perfect-square factor of 72 is 36: √72 = √(36·2) = √36·√2 = 6√2.

Worked example. Simplify √50 + √18. Simplify each: √50 = √(25·2) = 5√2 and √18 = √(9·2) = 3√2. Now they are like radicals: 5√2 + 3√2 = 8√2.

For higher roots, pull out perfect nth powers. Cube root: ³√54 = ³√(27·2) = 3·³√2, because 27 = 3³.

Useful products: √a·√b = √(ab) and √a/√b = √(a/b). So √2·√8 = √16 = 4.

Worked example — variables. Simplify √(x⁵). Write x⁵ = x⁴·x, where x⁴ is a perfect square: √(x⁴·x) = x²√x (for x ≥ 0). The rule is that even exponents come straight out (halved), and any leftover odd factor stays under the radical.

Worked example — multiply then simplify. Simplify √6·√10. Combine first: √60 = √(4·15) = 2√15. Combining under one radical before factoring often reveals a perfect square you would miss otherwise.

Common mistake: √(a + b) is NOT √a + √b. For instance √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The root does not distribute over addition.

Rational (Fractional) Exponents

A rational exponent is just another way to write a radical. The rule is:

x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ

The denominator is the root (index) and the numerator is the power. A negative sign on the exponent still means reciprocal.

Worked example. Evaluate 8^(2/3). Take the cube root first (smaller numbers): ³√8 = 2, then square: 2² = 4. Doing the root before the power keeps the arithmetic easy.

Worked example. Simplify 16^(3/4). The 4th root of 16 is 2 (since 2⁴ = 16), then cube: 2³ = 8.

Rational exponents obey all the normal exponent rules, so x^(1/2)·x^(1/3) = x^(1/2 + 1/3) = x^(5/6).

Worked example — negative rational exponent. Evaluate 4^(−3/2). The negative sign means reciprocal, so 4^(−3/2) = 1/4^(3/2) = 1/(√4)³ = 1/2³ = 1/8.

Adding and subtracting radicals works like combining like terms: only like radicals (same index and same radicand) combine. So 7√5 − 2√5 = 5√5, but 7√5 − 2√3 stays as is — they are unlike. Always simplify each radical first, because √20 + √45 looks unlike until you reduce: √20 = 2√5 and √45 = 3√5, giving 2√5 + 3√5 = 5√5.

Rationalizing Denominators

Convention says a simplified expression has no radical in the denominator. Clear it by multiplying top and bottom by the right factor.

Single-term denominator. Simplify 6/√3. Multiply by √3/√3: (6√3)/(√3·√3) = (6√3)/3 = 2√3.

Two-term denominator — use the conjugate. The conjugate of (a + √b) is (a − √b); multiplying them gives a² − b (no radical) via the difference-of-squares pattern (a+√b)(a−√b) = a² − b. Simplify 4/(3 − √5): multiply by (3 + √5)/(3 + √5). The denominator becomes 3² − (√5)² = 9 − 5 = 4, so the result is 4(3 + √5)/4 = 3 + √5.

Worked example — conjugate with a radical numerator. Rationalize 2/(√7 + √3). Multiply by the conjugate (√7 − √3): the denominator becomes (√7)² − (√3)² = 7 − 3 = 4, giving 2(√7 − √3)/4 = (√7 − √3)/2.

Complex Numbers

Some equations have no real solution — like x² = −1. To handle them we define the imaginary unit i by i² = −1, so i = √(−1). A complex number has the form a + bi, where a is the real part and b is the imaginary part.

Powers of i cycle every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then repeat.

Add/subtract componentwise: (3 + 2i) + (1 − 5i) = (3 + 1) + (2 − 5)i = 4 − 3i.

Multiply with FOIL, then replace i² with −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i − 3(−1) = 2 + i + 3 = 5 + i.

Simplify a root of a negative. √(−16) = √16·√(−1) = 4i. And √(−9)·√(−4) must be done by converting first: (3i)(2i) = 6i² = −6 (a common trap if you wrongly write √36 = 6).

Worked example — powers of i. Simplify i²³. Divide the exponent by 4 and keep the remainder: 23 = 4·5 + 3, so i²³ = i³ = −i. The remainder tells you where in the four-step cycle (i, −1, −i, 1) you land.

Worked example — subtract. (7 − 2i) − (4 + 6i) = (7 − 4) + (−2 − 6)i = 3 − 8i. Distribute the minus sign to both parts of the second number before combining.

Why do complex numbers matter for placement? They appear when the quadratic formula produces a negative under the square root (a negative discriminant b² − 4ac), signaling that the parabola never crosses the x-axis — the two roots are a complex-conjugate pair a ± bi.

Common mistake: √(−a)·√(−b) ≠ √(ab). Convert each to the form bi before multiplying, then apply i² = −1.