7-5 Operations with Radical Expressions

Property

Like radicals are radical expressions with the same radicand (the expression under the radical sign). 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. Sometimes you must first simplify radicals to see if they are like.

Examples

Property

If $a, b$ are nonnegative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ and $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
To multiply radicals with coefficients, multiply the coefficients together and then the variables, just like multiplying algebraic terms.

Examples

• To simplify $\sqrt{3} \cdot \sqrt{6}$, we multiply to get $\sqrt{18}$. Then, we simplify $\sqrt{18}$ as $\sqrt{9 \cdot 2}$, which results in $3\sqrt{2}$.
• For $(5\sqrt{2})(4\sqrt{8})$, multiply the coefficients $(5 \cdot 4)$ and the radicals $(\sqrt{2} \cdot \sqrt{8})$. This gives $20\sqrt{16}$, which simplifies to $20 \cdot 4 = 80$.
• To simplify $(\sqrt{10x})(\sqrt{5x^3})$, multiply under the radical to get $\sqrt{50x^4}$. This simplifies to $\sqrt{25x^4 \cdot 2}$, which is $5x^2\sqrt{2}$.

Explanation

This property allows you to combine two separate square roots into one by multiplying the numbers under the radicals. Afterwards, always check if the new radical can be simplified by factoring out any perfect squares.

Property

To multiply radical expressions with multiple terms, use the Distributive Property. Distribute each term in the first expression to each term in the second expression, then simplify any resulting radicals.

Examples