Operations with radicals

When you reach for it

Square roots turn up in the middle of real calculations. A rectangle with sides \displaystyle \sqrt{14} and \displaystyle \sqrt{10} has an area, a right triangle whose legs are roots has a perimeter, and one length can be divided by another. Each of these is an expression with several roots in it, and the aim is to write it as simply as possible while keeping it exact, never a rounded decimal.

Simplifying a square root already gives one root its simplest form: \displaystyle \sqrt{90} = 3\sqrt{10}. This lesson starts from there and combines roots with each other. There are three moves. Roots multiply under one root sign, they divide under one root sign, and they add only when they are multiples of the same root. After any of the three, the result is simplified again.

The procedure

Multiplying square roots

For any numbers a \ge 0 and b \ge 0,

This is the product rule from Simplifying a square root, read from right to left. There it took a root apart; here it puts two roots together.

Why it holds. The left side is a product of two non-negative numbers, so it is non-negative. Squaring it, and regrouping the factors (the commutative and associative properties of multiplication),

The right side, \displaystyle \sqrt{ab}, is by definition the non-negative number whose square is ab. Two non-negative numbers with the same square are the same number, so the two sides are equal.

A root squared. When a = b the rule gives \displaystyle \sqrt{a}\cdot\sqrt{a} = \sqrt{a^2} = a, so

which is simply what \displaystyle \sqrt{a} means: the number whose square is a. So \displaystyle \left(\sqrt{11}\right)^2 = 11.

Numbers in front. In \displaystyle 3\sqrt{2} the 3 is the coefficient and 2 is the radicand, the number under the root. To multiply \displaystyle m\sqrt{a} by \displaystyle n\sqrt{b}, reorder and regroup the four factors, exactly as 4x \cdot 3y = 12xy:

The product of two roots in simplest form is often not in simplest form itself: \displaystyle \sqrt{6}\cdot\sqrt{15} = \sqrt{90}, and \displaystyle \sqrt{90} = 3\sqrt{10}. So every product ends with a look for perfect square factors under the root.

Distributing over sums

An expression such as \displaystyle 3 + \sqrt{6} is a sum of two terms, so it multiplies like any binomial. The distributive property sends a single factor to every term, and two binomials multiply term by term, as in Multiplying two binomials. The only new step is that each product of two roots is then simplified, and \displaystyle \left(\sqrt{a}\right)^2 becomes a.

One pattern deserves a name. From Special products, (p + q)(p - q) = p^2 - q^2, because the two middle terms cancel. Put a root in place of p:

The root is squared, so it disappears, and the product has no root left in it. The two factors \displaystyle \sqrt{a} + q and \displaystyle \sqrt{a} - q are called conjugates.

Dividing square roots

For any numbers a \ge 0 and b > 0,

Why it holds. Both sides are non-negative. The square of the left side is \displaystyle \frac{\left(\sqrt{a}\right)^2}{\left(\sqrt{b}\right)^2} = \frac{a}{b}, and the square of the right side is \displaystyle \frac{a}{b} by definition. Same square, both non-negative, so they are equal. The condition b > 0 is there because \displaystyle \sqrt{0} = 0, and nothing can be divided by zero.

The rule is used in both directions. Reading left to right puts a quotient under one root, which pays off when b divides a: \displaystyle \frac{\sqrt{80}}{\sqrt{5}} = \sqrt{16} = 4. Reading right to left splits a fraction under a root into two roots, which pays off when the bottom is a perfect square: \displaystyle \sqrt{\frac{7}{9}} = \frac{\sqrt{7}}{3}. Coefficients divide on their own, as \displaystyle \frac{6x}{2y} = \frac{3x}{y}.

Sometimes a root is left below the line, as in \displaystyle \frac{\sqrt{7}}{\sqrt{3}}. That is a correct, exact value; turning it into a form with no root in the denominator is rationalizing the denominator, which comes later.

Adding and subtracting

Two roots are like radicals when, once each is in simplest form, they have the same radicand. \displaystyle 3\sqrt{2} and \displaystyle 5\sqrt{2} are like radicals; \displaystyle 4\sqrt{3} and \displaystyle 2\sqrt{5} are not.

Like radicals combine exactly as like terms do, by the distributive property:

It is 3x + 5x = 8x with \displaystyle \sqrt{2} in place of x. Subtraction works the same way, with the sign traveling with its term, as in Combining like terms.

Roots that look unlike may turn out to be like ones, which is why each root is simplified first: \displaystyle \sqrt{8} = 2\sqrt{2}, so \displaystyle \sqrt{8} and \displaystyle \sqrt{2} are like radicals.

Unlike radicals do not combine. \displaystyle 4\sqrt{3} + 2\sqrt{5} is already in simplest form, and its value shows that no single multiple of one root replaces it:

while \displaystyle 6\sqrt{3} \approx 10.392 and \displaystyle 6\sqrt{5} \approx 13.416. A sum of two roots is also not the root of the sum; Simplifying a square root shows why there is no “sum rule”.

Worked examples

Boundaries

• Both rules need the roots to be real. \displaystyle \sqrt{a}\cdot\sqrt{b} = \sqrt{ab} is claimed only for a \ge 0 and b \ge 0; with a = -4 and b = -9 the right side is \displaystyle \sqrt{36} = 6, but the left side is not a real number at all.
• The quotient rule needs b > 0. A zero under the bottom root means dividing by zero.
• Only like radicals add. Adding or subtracting changes the coefficient and never the radicand. Unlike radicals, such as \displaystyle \sqrt{2} and \displaystyle \sqrt{3}, stay as two terms.
• Same kind of root. Cube roots obey the same product rule among themselves, \displaystyle \sqrt[3]{2}\cdot\sqrt[3]{4} = \sqrt[3]{8} = 2, but these rules never join a square root to a cube root.

Where this goes next

The conjugate product in Example 3 is the key to Rationalizing a denominator: multiplying \displaystyle \frac{1}{\sqrt{13} + 3} top and bottom by \displaystyle \sqrt{13} - 3 turns the bottom into 4. Answers from the quadratic formula arrive as expressions like \displaystyle \frac{6 \pm \sqrt{20}}{2}, which these rules reduce to \displaystyle 3 \pm \sqrt{5}. Lengths from the The distance formula and the side ratios in the special right triangles are roots that get multiplied, divided and added in exactly this way. Underneath it all, Irrational numbers explains why these values are kept as roots rather than rounded.

Exercises

Multiplying

1. \displaystyle \sqrt{2}\cdot\sqrt{98}
2. \displaystyle \sqrt{22}\cdot\sqrt{55}
3. \displaystyle 5\sqrt{6}\cdot 2\sqrt{33}
4. \displaystyle \left(3\sqrt{5}\right)^2

Distributing and expanding

5. \displaystyle \sqrt{3}\left(\sqrt{15} + 2\right)
6. \displaystyle \left(4 + \sqrt{3}\right)\left(1 + 2\sqrt{3}\right)
7. \displaystyle \left(\sqrt{10} - 3\right)^2
8. \displaystyle \left(\sqrt{17} + 4\right)\left(\sqrt{17} - 4\right)
9. \displaystyle \left(3\sqrt{2} + \sqrt{7}\right)\left(3\sqrt{2} - \sqrt{7}\right)

Dividing

10. \displaystyle \dfrac{\sqrt{468}}{\sqrt{13}}
11. \displaystyle \dfrac{20\sqrt{78}}{4\sqrt{6}}
12. \displaystyle \sqrt{\dfrac{50}{9}}

Adding and subtracting

13. \displaystyle \sqrt{68} + \sqrt{153} - \sqrt{272}
14. \displaystyle 2\sqrt{63} - \sqrt{28} + \sqrt{18}
15. \displaystyle 3\sqrt{20} - 2\sqrt{125}

Problems

16. A right triangle has legs \displaystyle \sqrt{117} cm and \displaystyle \sqrt{208} cm. Find its hypotenuse and its perimeter exactly, and the perimeter to two decimal places.
17. A rectangular mural has an area of \displaystyle 12\sqrt{10} square meters and a width of \displaystyle 2\sqrt{5} meters. How tall is it?
18. \displaystyle \sqrt{108} and \displaystyle \sqrt{3} have different radicands. Find \displaystyle \sqrt{108} - \sqrt{3}, and say why the two could be combined.

Reasoning

19. Show that \displaystyle \left(\sqrt{a} + q\right)\left(\sqrt{a} - q\right) = a - q^2 for any a \ge 0, and explain why the product of a pair of conjugates has no root in it.