Simplify Square Roots

By the end of this section, you will be able to: use the Product Property to simplify square roots, and use the Quotient Property to simplify square roots.

In the last section we estimated the square root of a number between two consecutive whole numbers. We could say that $\sqrt{50}$ is between $7$ and $8$ . This is easy to do when the numbers are small enough that we can estimate them, but what if we want to estimate $\sqrt{500}$ ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots, as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

Simplified Square Root.

\sqrt{a}

is considered simplified if

a

has no perfect square factors.

So $\sqrt{31}$ is simplified. But $\sqrt{32}$ is not simplified, because $16$ is a perfect square factor of $32$ .

Use the Product Property to simplify square roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that $(ab)^m = a^m b^m$ . The corresponding property of square roots says that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ .

Product Property of Square Roots. If $a$ , $b$ are non-negative real numbers, then

\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}

We use the Product Property of Square Roots to remove all perfect square factors from a radical.

Example. Simplify $\sqrt{50}$ .

Find the largest perfect square factor of the radicand, rewrite the radicand as a product using that factor, split the radical, and simplify the square root of the perfect square:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{50} &=& \sqrt{25 \cdot 2} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{25} \cdot \sqrt{2} \\[4pt] \text{Simplify.} && & 5\sqrt{2} \end{array}

Notice that the simplified form of $\sqrt{50}$ is $5\sqrt{2}$ , which is the product of an integer and a square root. We always write the integer in front of the square root.

How to simplify a square root using the Product Property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using that perfect square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

Simplify: $\sqrt{48}$ .

Example. Simplify $\sqrt{500}$ .

The largest perfect square factor of $500$ is $100$ :

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{500} &=& \sqrt{100 \cdot 5} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{100} \cdot \sqrt{5} \\[4pt] \text{Simplify.} && & 10\sqrt{5} \end{array}

We could use the simplified form $10\sqrt{5}$ to estimate $\sqrt{500}$ . We know $5$ is between $2$ and $3$ , so $\sqrt{500}$ is between $20$ and $30$ .

The next example is much like the previous ones, but with variables.

Example. Simplify $\sqrt{x^3}$ .

Rewrite $x^3$ using its largest perfect square factor $x^2$ , split the radical, and simplify:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{x^3} &=& \sqrt{x^2 \cdot x} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{x^2} \cdot \sqrt{x} \\[4pt] \text{Simplify.} && & x\sqrt{x} \end{array}

Simplify: $\sqrt{b^5}$ .

We follow the same procedure when there is a coefficient in the radical, too.

Example. Simplify $\sqrt{25y^5}$ .

The largest perfect square factor of $25y^5$ is $25y^4$ :

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{25y^5} &=& \sqrt{25y^4 \cdot y} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{25y^4} \cdot \sqrt{y} \\[4pt] \text{Simplify.} && & 5y^2\sqrt{y} \end{array}

Simplify: $\sqrt{16x^7}$ .

In the next example both the constant and the variable have perfect square factors.

Example. Simplify $\sqrt{72n^7}$ .

The largest perfect square factor of $72n^7$ is $36n^6$ , leaving a factor of $2n$ inside the radical:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{72n^7} &=& \sqrt{36n^6 \cdot 2n} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{36n^6} \cdot \sqrt{2n} \\[4pt] \text{Simplify.} && & 6n^3\sqrt{2n} \end{array}

Simplify: $\sqrt{32y^5}$ .

Example. Simplify $\sqrt{63u^3 v^5}$ .

The largest perfect square factor is $9u^2 v^4$ , leaving $7uv$ inside the radical:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \sqrt{63u^3 v^5} &=& \sqrt{9u^2 v^4 \cdot 7uv} \\[4pt] \text{Rewrite as the product of two radicals.} && & \sqrt{9u^2 v^4} \cdot \sqrt{7uv} \\[4pt] \text{Simplify.} && & 3uv^2\sqrt{7uv} \end{array}

Simplify: $\sqrt{98a^7 b^5}$ .

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify $\sqrt{25} + \sqrt{144}$ we must simplify each square root separately first, then add to get the sum of $17$ . The expression $\sqrt{17} + \sqrt{7}$ cannot be simplified — to begin we’d need to simplify each square root, but neither $17$ nor $7$ contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

Example. Simplify $3 + \sqrt{32}$ .

Simplify the square root; the terms are not like, so we cannot combine them:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & 3 + \sqrt{32} &=& 3 + \sqrt{16 \cdot 2} \\[4pt] \text{Rewrite as the product of two radicals.} && & 3 + \sqrt{16} \cdot \sqrt{2} \\[4pt] \text{Simplify.} && & 3 + 4\sqrt{2} \end{array}

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable — they are not like terms!

Simplify: $5 + \sqrt{75}$ .

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

Example. Simplify $\tfrac{4 - \sqrt{48}}{2}$ .

Simplify the radical, factor the common factor out of the numerator, and then remove the common factor of $2$ from the numerator and denominator:

\begin{array}{lrcl} \text{Rewrite using the largest perfect square factor.} & \tfrac{4 - \sqrt{48}}{2} &=& \tfrac{4 - \sqrt{16 \cdot 3}}{2} \\[10pt] \text{Rewrite as the product of two radicals.} && & \tfrac{4 - \sqrt{16} \cdot \sqrt{3}}{2} \\[10pt] \text{Simplify the radical.} && & \tfrac{4 - 4\sqrt{3}}{2} \\[10pt] \text{Factor the common factor from the numerator.} && & \tfrac{4(1 - \sqrt{3})}{2} \\[10pt] \text{Remove the common factor } 2. && & 2(1 - \sqrt{3}) \end{array}

Simplify: $\tfrac{10 - \sqrt{75}}{5}$ .

Use the Quotient Property to simplify square roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Example. Simplify $\sqrt{\tfrac{9}{64}}$ .

Since $\left(\tfrac{3}{8}\right)^2 = \tfrac{9}{64}$ , the fraction is a perfect square, so

\sqrt{\tfrac{9}{64}} = \tfrac{3}{8}.

Simplify: $\sqrt{\tfrac{25}{16}}$ .

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Example. Simplify $\sqrt{\tfrac{45}{80}}$ .

Simplify the fraction inside the radical first by removing common factors, then take the square root of the perfect square fraction:

\begin{array}{lrcl} \text{Rewrite showing the common factors.} & \sqrt{\tfrac{45}{80}} &=& \sqrt{\tfrac{5 \cdot 9}{5 \cdot 16}} \\[10pt] \text{Simplify the fraction by removing common factors.} && & \sqrt{\tfrac{9}{16}} \\[10pt] \text{Simplify.} && & \tfrac{3}{4} \end{array}

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, $\tfrac{a^m}{a^n} = a^{m-n}$ , $a \neq 0$ .

Example. Simplify $\sqrt{\tfrac{m^6}{m^4}}$ .

Simplify the fraction inside the radical first by dividing the like bases, then take the square root:

\begin{array}{lrcl} \text{Divide the like bases by subtracting the exponents.} & \sqrt{\tfrac{m^6}{m^4}} &=& \sqrt{m^2} \\[4pt] \text{Simplify.} && & m \end{array}

Example. Simplify $\sqrt{\tfrac{48p^7}{3p^3}}$ .

Simplify the fraction inside the radical first, then take the square root of the perfect square:

\begin{array}{lrcl} \text{Simplify the fraction inside the radical first.} & \sqrt{\tfrac{48p^7}{3p^3}} &=& \sqrt{16p^4} \\[4pt] \text{Simplify.} && & 4p^2 \end{array}

Simplify: $\sqrt{\tfrac{75x^5}{3x}}$ .

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately: $\left(\tfrac{a}{b}\right)^m = \tfrac{a^m}{b^m}$ , $b \neq 0$ . We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Quotient Property of Square Roots. If $a$ , $b$ are non-negative real numbers and $b \neq 0$ , then

\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}

Example. Simplify $\sqrt{\tfrac{21}{64}}$ .

We cannot simplify the fraction inside the radical, so rewrite using the Quotient Property and simplify the square root of $64$ ; the numerator cannot be simplified:

\begin{array}{lrcl} \text{Rewrite using the Quotient Property.} & \sqrt{\tfrac{21}{64}} &=& \tfrac{\sqrt{21}}{\sqrt{64}} \\[10pt] \text{Simplify the square root of } 64. && & \tfrac{\sqrt{21}}{8} \end{array}

Simplify: $\sqrt{\tfrac{19}{49}}$ .

Example. Use the Quotient Property to simplify $\sqrt{\tfrac{27m^3}{196}}$ .

The fraction $\tfrac{27m^3}{196}$ cannot be simplified, so rewrite the radical as a quotient of two radicals, then simplify the radicals in the numerator and denominator ( $9m^2$ and $196$ are perfect squares):

\begin{array}{lrcl} \text{Rewrite using the Quotient Property.} & \sqrt{\tfrac{27m^3}{196}} &=& \tfrac{\sqrt{27m^3}}{\sqrt{196}} \\[10pt] \text{Simplify the radicals in the numerator and denominator.} && & \tfrac{\sqrt{9m^2} \cdot \sqrt{3m}}{\sqrt{196}} \\[10pt] \text{Simplify.} && & \tfrac{3m\sqrt{3m}}{14} \end{array}

How to simplify a square root using the Quotient Property.

Simplify the fraction in the radicand, if possible.
Use the Quotient Property to rewrite the radical as the quotient of two radicals.
Simplify the radicals in the numerator and the denominator.

Example. Simplify $\sqrt{\tfrac{45x^5}{y^4}}$ .

We cannot simplify the fraction in the radicand, so rewrite using the Quotient Property, then simplify the radicals in the numerator and denominator:

\begin{array}{lrcl} \text{Rewrite using the Quotient Property.} & \sqrt{\tfrac{45x^5}{y^4}} &=& \tfrac{\sqrt{45x^5}}{\sqrt{y^4}} \\[10pt] \text{Simplify the radicals in the numerator and denominator.} && & \tfrac{\sqrt{9x^4} \cdot \sqrt{5x}}{y^2} \\[10pt] \text{Simplify.} && & \tfrac{3x^2\sqrt{5x}}{y^2} \end{array}

Simplify: $\sqrt{\tfrac{80m^3}{n^6}}$ .

Be sure to simplify the fraction in the radicand first, if possible.

Example. Simplify $\sqrt{\tfrac{81d^9}{25d^4}}$ .

Simplify the fraction in the radicand first, then rewrite using the Quotient Property and simplify the radicals in the numerator and denominator:

\begin{array}{lrcl} \text{Simplify the fraction in the radicand.} & \sqrt{\tfrac{81d^9}{25d^4}} &=& \sqrt{\tfrac{81d^5}{25}} \\[10pt] \text{Rewrite using the Quotient Property.} && & \tfrac{\sqrt{81d^5}}{\sqrt{25}} \\[10pt] \text{Simplify the radicals in the numerator and denominator.} && & \tfrac{\sqrt{81d^4} \cdot \sqrt{d}}{5} \\[10pt] \text{Simplify.} && & \tfrac{9d^2\sqrt{d}}{5} \end{array}

Example. Simplify $\sqrt{\tfrac{18p^5 q^7}{32pq^2}}$ .

Simplify the fraction in the radicand, then rewrite using the Quotient Property and simplify each radical:

\begin{array}{lrcl} \text{Simplify the fraction in the radicand.} & \sqrt{\tfrac{18p^5 q^7}{32pq^2}} &=& \sqrt{\tfrac{9p^4 q^5}{16}} \\[10pt] \text{Rewrite using the Quotient Property.} && & \tfrac{\sqrt{9p^4 q^5}}{\sqrt{16}} \\[10pt] \text{Simplify the radicals in the numerator and denominator.} && & \tfrac{\sqrt{9p^4 q^4} \cdot \sqrt{q}}{4} \\[10pt] \text{Simplify.} && & \tfrac{3p^2 q^2\sqrt{q}}{4} \end{array}

Simplify: $\sqrt{\tfrac{50x^5 y^3}{72x^4 y}}$ .

Key terms

simplified square root — a square root whose radicand contains no perfect square factors. perfect square fraction — a fraction in which both the numerator and the denominator are perfect squares. Product Property of Square Roots — for non-negative $a$ and $b$ , $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ . Quotient Property of Square Roots — for non-negative $a$ and $b$ with $b \neq 0$ , $\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}$ .

This section is adapted from Elementary Algebra 2e, 9.2 Simplify 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.

Simplify and Use Square Roots Add and Subtract Square Roots