Lesson 8.4: Add, Subtract, and Multiply Radical Expressions

New Concept

This lesson expands your algebra toolkit to radical expressions. You'll master adding, subtracting, and multiplying radicals by treating them like polynomials—combining like radicals and applying familiar multiplication properties to simplify expressions.

What’s next

Soon, you'll tackle practice cards on combining like radicals and watch short videos that break down multiplying radical expressions.

Property

Like radicals are radical expressions with the same index and the same radicand. We add and subtract like radicals in the same way we add and subtract like terms. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms. For radicals to be like, they must have the same index and radicand. Sometimes you must first simplify radicals to see if they are like.

Examples

• Simplify $8\sqrt{3} - 2\sqrt{3}$. Since the radicals are like, we subtract the coefficients: $(8-2)\sqrt{3} = 6\sqrt{3}$.

• Simplify $\sqrt{27} + 5\sqrt{3}$. First, simplify $\sqrt{27}$ to $\sqrt{9 \cdot 3} = 3\sqrt{3}$. Now combine like radicals: $3\sqrt{3} + 5\sqrt{3} = 8\sqrt{3}$.

Property

For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b}$, and for any integer $n \geq 2$:

Examples

• Simplify $(4\sqrt{3})(2\sqrt{5})$. Multiply the coefficients and the radicands: $(4 \cdot 2)\sqrt{3 \cdot 5} = 8\sqrt{15}$.

• Simplify $(-3\sqrt[3]{4})(5\sqrt[3]{4})$. Multiply to get $-15\sqrt[3]{16}$. Then simplify: $-15\sqrt[3]{8 \cdot 2} = -15 \cdot 2\sqrt[3]{2} = -30\sqrt[3]{2}$.

Property

We use the Distributive Property to multiply expressions with radicals, just as we do with polynomials. When multiplying binomials containing radicals, the FOIL (First, Outer, Inner, Last) method helps ensure all four products are found. After multiplying, combine any like terms.

Examples

• Simplify $\sqrt{5}(3 + \sqrt{10})$. Distribute $\sqrt{5}$: $3\sqrt{5} + \sqrt{50}$. Simplify the second term: $3\sqrt{5} + \sqrt{25 \cdot 2} = 3\sqrt{5} + 5\sqrt{2}$.

• Simplify $(4 - 2\sqrt{3})(5 + \sqrt{3})$. Using FOIL: $4(5) + 4(\sqrt{3}) - 2\sqrt{3}(5) - 2\sqrt{3}(\sqrt{3}) = 20 + 4\sqrt{3} - 10\sqrt{3} - 6 = 14 - 6\sqrt{3}$.

Property

Recognizing special product patterns from polynomials makes multiplying radicals easier.

Binomial Squares