Multiply Square Roots

Multiply square roots

We have used the Product Property of Square Roots to simplify square roots by removing perfect square factors. The Product Property of Square Roots says

We can use the Product Property of Square Roots “in reverse” to multiply square roots:

Remember, we assume all variables are greater than or equal to zero. We will rewrite the Product Property of Square Roots so we see both ways together.

So we can multiply $\sqrt{3} \cdot \sqrt{5}$ this way:

Sometimes the product gives us a perfect square:

Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply $4x \cdot 3y$ we multiply the coefficients together and then the variables. The result is $12xy$. Keep this in mind as you do these examples.

Example. Simplify: (a) $\sqrt{2} \cdot \sqrt{6}$ and (b) $\left(4\sqrt{3}\right)\left(2\sqrt{12}\right)$.

(a) Multiply using the Product Property, then simplify the radical:

(b) Multiply the coefficients and multiply the radicals, then simplify:

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify $\sqrt{12}$. We waited to get the product and then simplified.

When we have to multiply square roots, we first find the product and then remove any perfect square factors.

Example. Simplify: $\left(6\sqrt{2}\right)\left(3\sqrt{10}\right)$.

Multiply the coefficients and the radicals, then simplify:

Example. Simplify: (a) $\left(\sqrt{8x^3}\right)\left(\sqrt{3x}\right)$ and (b) $\left(\sqrt{20y^2}\right)\left(\sqrt{5y^3}\right)$.

(a) Multiply, then simplify the radical:

(b) Multiply, then simplify the radical:

Example. Simplify: $\left(10\sqrt{6p^3}\right)\left(3\sqrt{18p}\right)$.

Multiply the coefficients and the radicals, then simplify:

When we multiply two like square roots, it is the same as squaring a single square root.

Example. Simplify: (a) $\left(\sqrt{2}\right)^2$ and (b) $\left(-\sqrt{11}\right)^2$.

(a) Rewrite as a product, multiply, and simplify:

(b) Rewrite as a product, multiply, and simplify:

The results of the previous example lead us to this property.

By realizing that squaring and taking a square root are “opposite” operations, we can simplify $\left(\sqrt{2}\right)^2$ and get $2$ right away.

Example. Simplify: (a) $\left(2\sqrt{3}\right)\left(8\sqrt{3}\right)$ and (b) $\left(3\sqrt{6}\right)^2$.

(a) Multiply, using $\left(\sqrt{3}\right)^2 = 3$:

(b) Square the coefficient and the radical:

Use polynomial multiplication to multiply square roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots. We will first distribute and then simplify the square roots when possible.

Example. Simplify: (a) $3\left(5 - \sqrt{2}\right)$ and (b) $\sqrt{2}\left(4 - \sqrt{10}\right)$.

(a) Distribute:

(b) Distribute, then simplify the radical:

Example. Simplify: (a) $\sqrt{5}\left(7 + 2\sqrt{5}\right)$ and (b) $\sqrt{6}\left(\sqrt{2} + \sqrt{18}\right)$.

(a) Distribute, using $\left(\sqrt{5}\right)^2 = 5$, then simplify:

(b) Distribute, simplify each radical, then combine like radicals:

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work — usually by the FOIL method.

Example. Simplify: $\left(2 + \sqrt{3}\right)\left(4 - \sqrt{3}\right)$.

Multiply the four products with FOIL, then combine like terms:

Example. Simplify: $\left(3 - 2\sqrt{7}\right)\left(4 - 2\sqrt{7}\right)$.

Multiply the four products with FOIL, remembering $\sqrt{7} \cdot \sqrt{7} = 7$, then combine like terms:

Example. Simplify: $\left(3\sqrt{2} - \sqrt{5}\right)\left(\sqrt{2} + 4\sqrt{5}\right)$.

Multiply the four products with FOIL, then combine like terms:

Example. Simplify: $\left(4 - 2\sqrt{x}\right)\left(1 + 3\sqrt{x}\right)$.

Multiply the four products with FOIL, remembering $\sqrt{x} \cdot \sqrt{x} = x$, then combine like terms:

Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.

Example. Simplify: (a) $\left(2 + \sqrt{3}\right)^2$ and (b) $\left(4 - 2\sqrt{5}\right)^2$.

Be sure to include the $2ab$ term when squaring a binomial.

(a) Use the binomial square pattern $(a + b)^2 = a^2 + 2ab + b^2$:

(b) Use the binomial square pattern $(a - b)^2 = a^2 - 2ab + b^2$:

Example. Simplify: $\left(1 + 3\sqrt{x}\right)^2$.

Use the binomial square pattern $(a + b)^2 = a^2 + 2ab + b^2$:

In the next two examples, we will find the product of conjugates.

Example. Simplify: $\left(4 - \sqrt{2}\right)\left(4 + \sqrt{2}\right)$.

Use the product-of-conjugates pattern $(a - b)(a + b) = a^2 - b^2$:

Example. Simplify: $\left(5 - 2\sqrt{3}\right)\left(5 + 2\sqrt{3}\right)$.

Use the product-of-conjugates pattern $(a - b)(a + b) = a^2 - b^2$:

Notice that the product of conjugates always gives a result with no radical — the middle terms of the FOIL cancel and the outer squares leave a rational number. This idea, expressed by $\left(\sqrt{a} + \sqrt{b}\right)\left(\sqrt{a} - \sqrt{b}\right) = a - b$, becomes very useful in the next section when we divide square roots.

Key terms

Product Property of Square Roots — for nonnegative $a$ and $b$, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$; used “in reverse” to multiply two square roots into one. squaring a square root — for nonnegative $a$, $\left(\sqrt{a}\right)^2 = a$, since squaring undoes a square root. conjugates — a pair of binomials that differ only in the sign between their terms, such as $a + \sqrt{b}$ and $a - \sqrt{b}$; their product $a^2 - b$ contains no radical.

This section is adapted from Elementary Algebra 2e, 9.4 Multiply Square Roots by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked “How To” step tables as display equality chains with left-hand explanations; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.