5.2 Radical expressions & equations

Simplifying Radicals

A radical expression contains a root symbol, most often a square root such as the square root of 50. To simplify, use the product rule: the square root of (a times b) equals (root a)(root b). The goal is to pull out perfect-square factors so nothing remaining under the root is a perfect square.

Example 1. Simplify root 50. Since 50 = 25 times 2 and 25 is a perfect square, root 50 = (root 25)(root 2) = 5 root 2.

Example 2. Simplify root 72. Factor 72 = 36 times 2, so root 72 = (root 36)(root 2) = 6 root 2. Choosing the largest perfect-square factor finishes in one step; if you use 9 times 8 you must simplify again.

For variables, root(x^8) = x^4 because (x^4)^2 = x^8. For odd powers, root(x^7) = root(x^6 times x) = x^3 root x.

Quotient Rule

The square root of (a/b) equals (root a)/(root b). So root(9/16) = (root 9)/(root 16) = 3/4. This lets you split a fraction sitting under one radical into two simpler roots you can evaluate separately.

Rationalizing Denominators

Standard form keeps radicals out of the denominator. Multiply the fraction by a form of 1 that clears the root.

Example 3. Rationalize 3/(root 5). Multiply top and bottom by root 5: (3 times root 5)/((root 5)(root 5)) = (3 root 5)/5.

Example 4 (binomial denominator). Rationalize 4/(3 - root 2). Multiply by the conjugate 3 + root 2: 4(3 + root 2) / [(3 - root 2)(3 + root 2)]. The denominator becomes a difference of squares: 3^2 - (root 2)^2 = 9 - 2 = 7. The result is (12 + 4 root 2)/7.

Operations With Radicals

Adding and subtracting: you may only combine like radicals — the same index and the same radicand — the way you combine like terms.

Example 5. Simplify root 18 + root 8. Simplify each term first: root 18 = 3 root 2 and root 8 = 2 root 2. Now 3 root 2 + 2 root 2 = 5 root 2. You could not combine them before simplifying, because they did not look alike yet.

Multiplying: multiply the coefficients and the radicands separately, then always simplify the final radical, because a perfect square hidden inside the product is easy to miss. For example, 2 root 3 times 5 root 6 = 10 root 18 = 10 times 3 root 2 = 30 root 2.

Rational (Fractional) Exponents

A fractional exponent is another way to write a root: a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a to the m power, which also equals (nth root of a) to the m power. The denominator is the root and the numerator is the power.

• 16^(1/2) = root 16 = 4
• 8^(1/3) = cube root of 8 = 2
• 27^(2/3) = (cube root of 27)^2 = 3^2 = 9
• 16^(3/4) = (fourth root of 16)^3 = 2^3 = 8

Fractional exponents obey all the usual exponent rules. For instance, x^(1/2) times x^(1/3) = x^(1/2 + 1/3) = x^(5/6), and (9x^4)^(1/2) = 3x^2.

Solving Radical Equations

Method: isolate the radical on one side, then square both sides to remove a square root. Solve the resulting equation, then check every answer — squaring can introduce extraneous roots that do not satisfy the original equation.

Example 6. Solve root(x + 6) = 4. The radical is already isolated, so square both sides: x + 6 = 16, giving x = 10. Check: root(10 + 6) = root 16 = 4. The answer is valid.

Example 7 (isolate first). Solve root(2x - 1) + 3 = 8. Isolate the radical: root(2x - 1) = 5. Square both sides: 2x - 1 = 25, so 2x = 26 and x = 13. Check: root(26 - 1) + 3 = root 25 + 3 = 5 + 3 = 8. The answer holds.

Example 8 (extraneous). Solve root(x + 3) = x - 3. Square both sides: x + 3 = (x - 3)^2 = x^2 - 6x + 9. Move everything to one side: 0 = x^2 - 7x + 6 = (x - 1)(x - 6). The candidates are x = 1 and x = 6. Check x = 6: root 9 = 3 and 6 - 3 = 3, so it works. Check x = 1: root 4 = 2 but 1 - 3 = -2, which fails. So x = 1 is extraneous, and the only solution is x = 6.

Quick Reference

Squaring is the exact step that creates extraneous roots, so a checked answer is the only safe answer on the PERT.