Topics
5. Exponents, Radicals, & Exponential Functions
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Algebra 1

5.06 Simplify radicals

Lesson
Practice

Simplify square and cube roots

Radical expressions have many parts as shown in the following diagram:

A diagram showing the parts of a radical with cube root of 27 as an example. The radical symbol is drawn like a check mark and long division symbol combined. The number 27 is beneath the horizontal bar of a radical symbol and is labeled as the radicand. The small number 3 tucked outside the radical symbol and is labeled as the index. The whole thing is labeled as the Radical.
Index

The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. The index on a square root is usually not written, but would be 2

Radical

A mathematical expression that uses a root, such as a square root \sqrt{\quad}, or nth root \sqrt[n]{\quad}

Radicand

The value or expression inside the radical symbol

Perfect square

A number that is the result of multiplying two of the same integer

Perfect cube

A number that is the result of multiplying three of the same integer together

Radical expressions are written in simplified radical form if the radicand cannot be factored any further. For square roots, this means there are no remaining factors of the radicand that are perfect squares and for cube roots this means there are no remaining factors of the radicand that are perfect cubes. If an expression is in simplified radical form, and there is still a number left in the radicand, the result will be irrational.

We can use the following facts to simplify radical expressions, for a,\,b \geq 0 and m,\,n positive integers,\begin{aligned} \sqrt[n]{ab} &=\sqrt[n]{a} \sqrt[n]{b} \\ \sqrt[n]{a^n}&=a \\ a^{mn}&=\left(a^m\right)^n \end{aligned}

Exploration

We can use prime factors to help us split a number into a product of a perfect square and a remainder.

If we wanted to simplify \sqrt{60}, we want to see if 60 has any factors which are perfect squares.

A factor tree for 60, with factors 2 and 30, further dividing 30 into 3 and 10, and 10 into 2 and 5.

Using the factor tree, we can see that 60=2^2\cdot 3\cdot 5, so the perfect square factor is 2^2=4, and the remainder is 15.

\begin{aligned}\sqrt{60}&=\sqrt{2 \cdot 30} \\ &= \sqrt{2 \cdot 3\cdot 10}\\ &= \sqrt{2\cdot 2\cdot 3\cdot 5}\\ &=\sqrt{2^2}\cdot \sqrt{3\cdot 5} \\ &= 2\sqrt{15}\end{aligned}

  1. How can you identify if a number has a perfect square factor after drawing a factor tree?
  2. Is it possible to get two different factor trees for the same number?
  3. Is it possible to get two different prime factorizations?
  4. Can every radical be simplified?
  5. How would you rewrite 180 as a product of a perfect square and a remainder?
  6. What would happen if we didn't use the largest perfect square?

Prime factorization method

A figure showing how to simplify square root of 24. First line: square root of 24 equals square root of 2 times 2 times 2 times 3; Second line: equals square root of 2 squared times 2 times 3; Third line: equals square root of 2 squared times square root of 2 times square root of 3; Fourth line: equals 2 times square root of 2 times 3; Fifth line: equals 2 times square root of 6.

We can use a few steps to help us simplify any radical:

  1. Find prime factorization of radicand
  2. Group factors in groups equal to index of radical expression
  3. Use multiplication property of radicals
  4. Simplify any rational factors

Perfect square method

A figure showing how to simplify square root of 24. First line: square root of 24 equals square root of 4 times 6; Second line: equals square root of 4 times square root of 6; Third line: equals 2 times square root of 6.

This is the quickest method for simplifying a radical.

  1. Find largest perfect square (or cube) factor of radicand
  2. Use multiplication property of radicals
  3. Use the properties of radicals to factor
  4. Simplify any perfect square (or cube) factors

Examples

Example 1

Simplify \sqrt{12}

Worked Solution
Create a strategy

Split 12 into its prime factors and use \sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}.

Apply the idea
\displaystyle \sqrt{12}\displaystyle =\displaystyle \sqrt{2\cdot2\cdot3}Find prime factorization of 12
\displaystyle \text{ }\displaystyle =\displaystyle \sqrt{2^2\cdot3}Rewrite using perfect square
\displaystyle \text{ }\displaystyle =\displaystyle \sqrt{2^2}\cdot\sqrt{3}Product of radicals property
\displaystyle \text{ }\displaystyle =\displaystyle 2\sqrt{3}Square root of a perfect square

Example 2

Simplify \sqrt[3]{-64}

Worked Solution
Create a strategy

Split -64 into factors. Notice the negative sign.

Apply the idea
\displaystyle \sqrt[3]{-64}\displaystyle =\displaystyle \sqrt[3]{-4 \cdot -4 \cdot -4}Factor radicand
\displaystyle =\displaystyle \sqrt[3]{(-4)^3}Rewrite using perfect cube
\displaystyle =\displaystyle -4Cube root of perfect cube
Reflect and check

Negative radicands can have rational cube roots. Can the same be said for square roots?

Example 3

Simplify the expression 3\sqrt{18}.

Worked Solution
Create a strategy

To simplify the expression, we can first simplify the square root and then multiply the result by the coefficient outside the root.

Apply the idea
\displaystyle 3\sqrt{18}\displaystyle =\displaystyle 3\sqrt{9 \cdot 2}Factor 18 into 9 \cdot 2
\displaystyle =\displaystyle 3 \cdot \sqrt{9} \cdot \sqrt{2}Use the property of square roots that \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\displaystyle =\displaystyle 3 \cdot 3 \cdot \sqrt{2}Find the square root of 9, which is 3
\displaystyle =\displaystyle 9\sqrt{2}Multiply the coefficients together

Therefore, the simplified form of 3\sqrt{18} is 9\sqrt{2}.

Reflect and check

Let's check our solution by substituting back into the original expression and evaluating both sides.

\displaystyle 3\sqrt{18}\displaystyle =\displaystyle 3 \cdot 4.24Substitute \sqrt{18} with its decimal approximation 4.24
\displaystyle =\displaystyle 12.72Evaluate
\displaystyle 9\sqrt{2}\displaystyle =\displaystyle 9 \cdot 1.41Substitute \sqrt{2} with its decimal approximation 1.41
\displaystyle =\displaystyle 12.69Evaluate

The values on both sides are approximately equal, confirming that our solution is correct. The slight difference is due to the rounding off of the square roots to two decimal places.

Idea summary

Radical expressions can be simplified if the radicand is divisible by any perfect squares (for square root) or perfect cubes (for cube root). If a number can be simplified to remove the radical, it is rational. Otherwise, it is irrational.

Outcomes

A.EO.4

The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.EO.4a

Simplify and determine equivalent radical expressions involving the square root of a whole number in simplest form.

A.EO.4b

Simplify and determine equivalent radical expressions involving the cube root of an integer.

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