CR.13: Operations with Square Roots

Learning Outcomes

  • Evaluate principal square roots.
  • Use the product rule to simplify square roots.
  • Use the quotient rule to simplify square roots.
  • Add and subtract square roots.
  • Rationalize denominators.

When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[/latex], the square root of [latex]16[/latex] is [latex]4[/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

In general terms, if [latex]a[/latex] is a positive real number, then the square root of [latex]a[/latex] is a number that, when multiplied by itself, gives [latex]a[/latex]. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals [latex]a[/latex]. The square root obtained using a calculator is the principal square root.

The principal square root of [latex]a[/latex] is written as [latex]\sqrt{a}[/latex]. The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.

A General Note: Principal Square Root

The principal square root of [latex]a[/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[/latex]. It is written as a radical expression, with a symbol called a radical over the term called the radicand: [latex]\sqrt{a}[/latex].

Q & A

Does [latex]\sqrt{25}=\pm 5[/latex]?

No. Although both [latex]{5}^{2}[/latex] and [latex]{\left(-5\right)}^{2}[/latex] are [latex]25[/latex], the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is [latex]\sqrt{25}=5[/latex].

Example: Evaluating Square Roots

Evaluate each expression.

  1. [latex]\sqrt{100}[/latex]
  2. [latex]\sqrt{\sqrt{16}}[/latex]
  3. [latex]\sqrt{25+144}[/latex]
  4. [latex]\sqrt{49}-\sqrt{81}[/latex]

Q & A

For [latex]\sqrt{25+144}[/latex], can we find the square roots before adding?

No. [latex]\sqrt{25}+\sqrt{144}=5+12=17[/latex]. This is not equivalent to [latex]\sqrt{25+144}=13[/latex]. The order of operations requires us to add the terms in the radicand before finding the square root.

Try It

Evaluate each expression.

  1. [latex]\sqrt{225}[/latex]
  2. [latex]\sqrt{\sqrt{81}}[/latex]
  3. [latex]\sqrt{25 - 9}[/latex]
  4. [latex]\sqrt{36}+\sqrt{121}[/latex]

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Use the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\sqrt{15}[/latex] as [latex]\sqrt{3}\cdot \sqrt{5}[/latex]. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

A General Note: The Product Rule for Simplifying Square Roots

If [latex]a[/latex] and [latex]b[/latex] are nonnegative, the square root of the product [latex]ab[/latex] is equal to the product of the square roots of [latex]a[/latex] and [latex]b[/latex].

[latex]\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex]

How To: Given a square root radical expression, use the product rule to simplify it.

  1. Factor any perfect squares from the radicand.
  2. Write the radical expression as a product of radical expressions.
  3. Simplify.

Recall Prime Factorization

It can be helpful, when simplifying square roots, to write the radicand as a product of primes in order to find perfect squares under the radical.

Example. [latex]\sqrt{288} \quad=\quad \sqrt{2\cdot3^2\cdot4^2} \quad=\quad 3\cdot4\cdot \sqrt{2} \quad=\quad 12\sqrt{2}.[/latex]

Example: Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

  1. [latex]\sqrt{300}[/latex]
  2. [latex]\sqrt{162{a}^{5}{b}^{4}}[/latex]

Try It

Simplify [latex]\sqrt{50{x}^{2}{y}^{3}z}[/latex].

https://ohm.lumenlearning.com/multiembedq.php?id=110285&theme=oea&iframe_resize_id=mom20

How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

  1. Express the product of multiple radical expressions as a single radical expression.
  2. Simplify.

Example: Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.

[latex]\sqrt{12}\cdot \sqrt{3}[/latex]

Try It

Simplify [latex]\sqrt{50x}\cdot \sqrt{2x}[/latex] assuming [latex]x>0[/latex].

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Example: Multiply radicals using the distributive property

Multiply and simplify the expression.

[latex](9-\sqrt{7})(6+\sqrt{7})[/latex]

Try It

Multiply and simplify [latex](1+\sqrt{5})(4-\sqrt{5})[/latex].

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Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\sqrt{\dfrac{5}{2}}[/latex] as [latex]\dfrac{\sqrt{5}}{\sqrt{2}}[/latex].

A General Note: The Quotient Rule for Simplifying Square Roots

The square root of the quotient [latex]\dfrac{a}{b}[/latex] is equal to the quotient of the square roots of [latex]a[/latex] and [latex]b[/latex], where [latex]b\ne 0[/latex].

[latex]\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}[/latex]

How To: Given a radical expression, use the quotient rule to simplify it.

  1. Write the radical expression as the quotient of two radical expressions.
  2. Simplify the numerator and denominator.

Recall Simplifying Fractions

To simplify fractions, find common factors in the numerator and denominator that cancel.

Example:      [latex]\dfrac{24}{32}\quad=\quad\dfrac{\cancel{2}\cdot\cancel{2}\cdot\cancel{2}\cdot3}{\cancel{2}\cdot\cancel{2}\cdot\cancel{2}\cdot2\cdot2}\quad=\quad\dfrac{3}{2\cdot2}\quad=\quad\dfrac{3}{4}[/latex]

Example: Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

[latex]\sqrt{\dfrac{5}{36}}[/latex]

Try It

Simplify [latex]\sqrt{\dfrac{2{x}^{2}}{9{y}^{4}}}[/latex].

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Example: Using the Quotient Rule to Simplify an Expression with Two Square Roots

Simplify the radical expression.

[latex]\dfrac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}[/latex]

Try It

Simplify [latex]\dfrac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}[/latex].

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In the following video you will see more examples of how to simplify radical expressions with variables.

Adding and Subtracting Radical Expressions

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\sqrt{2}[/latex] and [latex]3\sqrt{2}[/latex] is [latex]4\sqrt{2}[/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\sqrt{18}[/latex] can be written with a [latex]2[/latex] in the radicand, as [latex]3\sqrt{2}[/latex], so [latex]\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}[/latex].

How To: Given a radical expression requiring addition or subtraction of square roots, solve.

  1. Simplify each radical expression.
  2. Add or subtract expressions with equal radicands.

Example: Adding Square Roots

Add [latex]5\sqrt{12}+2\sqrt{3}[/latex].

Try It

Add [latex]\sqrt{5}+6\sqrt{20}[/latex].

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Watch this video to see more examples of adding roots.

Example: Subtracting Square Roots

Subtract [latex]20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}[/latex].

Try It

Subtract [latex]3\sqrt{80x}-4\sqrt{45x}[/latex].

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in the next video we show more examples of how to subtract radicals.

Rationalize Denominators

Recall the identity property of Multiplication

We leverage an important and useful identity in this section in a technique commonly used in college algebra:

rewriting an expression by multiplying it by a well-chosen form of the number 1.

Because the multiplicative identity states that [latex]a\cdot1=a[/latex], we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we rationalize the denominator.

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\sqrt{c}[/latex], multiply by [latex]\dfrac{\sqrt{c}}{\sqrt{c}}[/latex].

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\sqrt{c}[/latex], then the conjugate is [latex]a-b\sqrt{c}[/latex].

How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.

  1. Multiply the numerator and denominator by the radical in the denominator.
  2. Simplify.

Example: Rationalizing a Denominator Containing a Single Term

Write [latex]\dfrac{2\sqrt{3}}{3\sqrt{10}}[/latex] in simplest form.

Try It

Write [latex]\dfrac{12\sqrt{3}}{\sqrt{2}}[/latex] in simplest form.

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How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.

  1. Find the conjugate of the denominator.
  2. Multiply the numerator and denominator by the conjugate.
  3. Use the distributive property.
  4. Simplify.

Example: Rationalizing a Denominator Containing Two Terms

Write [latex]\dfrac{4}{1+\sqrt{5}}[/latex] in simplest form.

Try It

Write [latex]\dfrac{7}{2+\sqrt{3}}[/latex] in simplest form.

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