The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. If an index does not appear on the root, it is implied to 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

To simplify a radical expression, the radicand must be broken down into factors using the index as a guide for simplifying. We use properties to help us simplify, such as

\displaystyle \sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}

\bm{a,b}

are positive integers or variables

\bm{n}

is a positive integer

We can also use the property of taking the n^{th} root of an n^{th} power

\displaystyle \sqrt[n]{a^{n}}=a

\bm{a}

is a positive integer or variable

\bm{n}

is a positive integer

The index of a radical indicates the number of identical prime factors to look for to create a perfect n^{th}.

Square root expressions are written in simplest form if the radicand has no perfect square factors other than one. For cube roots, this means there are no remaining factors of the radicand that are perfect cubes.

The expression \sqrt[3]{54y^{4}} can be simplified through prime factorization.

\displaystyle \sqrt[3]{54y^{4}}	\displaystyle =	\displaystyle \sqrt[3]{2 \cdot 3 \cdot 3 \cdot 3 \cdot y \cdot y \cdot y \cdot y}
	\displaystyle =	\displaystyle \sqrt[3]{2 \cdot 3^{3} \cdot y^{3} \cdot y}
	\displaystyle =	\displaystyle \sqrt[3]{2 \cdot 3^{3} \cdot y^{3} \cdot y}
	\displaystyle =	\displaystyle \sqrt[3]{3^{3}} \cdot \sqrt[3]{y^{3}} \cdot \sqrt[3]{2\cdot y}
	\displaystyle =	\displaystyle 3y \sqrt[3]{2y}

Recall that the square root of a negative number does not result in a real number. When simplifying square roots with a negative radicand, we can take out a factor of \sqrt{-1}=i to get an imaginary number.

Examples

Example 1

Simplify -6\sqrt{12x^{5}y^{3}}. Assume all variables are positive.

Worked Solution

Create a strategy

Write out the prime factorization and use the properties to take square roots of perfect squares and simplify.

Apply the idea

\displaystyle -6\sqrt{12x^{5}y^{2}}	\displaystyle =	\displaystyle -6 \sqrt{2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y}	Find prime factorization
	\displaystyle =	\displaystyle -6 \sqrt{2^{2} \cdot 3 \cdot x^{4} \cdot x \cdot y^{2}}	Rewrite using perfect squares
	\displaystyle =	\displaystyle -6 \cdot \sqrt{2^{2}} \cdot \sqrt{x^{4}} \cdot \sqrt{y^{2}} \cdot \sqrt{3 \cdot x }	Product of radicals property
	\displaystyle =	\displaystyle -6 \cdot 2 \cdot x^{2} \cdot y \cdot \sqrt{3 \cdot x}	Square root of a perfect square
	\displaystyle =	\displaystyle -12x^{2}y \sqrt{3x}	Simplify

Reflect and check

The phrase "assume all variables are positive" allows us to evaluate an expression such as \sqrt{x^{2}}. Because the variable is positive, \sqrt{x^{2}}=x. However, if the variable was negative, then \sqrt{x^{2}}=-x. For example:

If y=1, the simplified expression would be -12x^{2}\sqrt{3x}
If y=-1, the simplified expression would be 12x^{2}\sqrt{3x}

These two results are not the same since 12x^{2}\sqrt{3x}\neq-12x^{2}\sqrt{3x}.

If the instructions for the problem do not specific whether the variables are positive or negative, we would need to use absolute value bars to account for both positive and negative results.

In this case, the absolute value notation of the simplified expression would be -12\left|x\right|^{2}\vert y \vert \sqrt{3x}. Because the square of any real number is always non-negative, x^{2} stays positive whether x is positive or negative, so the absolute value bars around x are not necessary. -12x^{2}\vert y \vert \sqrt{3x} And by the definition of absolute value, this expression is equivalent to the piecewise expression shown, which accounts for both positive and negative values \begin{cases} 12x^{2}y \sqrt{3x}, & y \lt 0 \\ -12x^{2}y \sqrt{3x}, & y\geq 0\end{cases}

Example 2

Simplify -a\sqrt[4]{80a^{4}}

Worked Solution

Create a strategy

Split 80 into its prime factors and use \sqrt[4]{a\cdot b}=\sqrt[4]{a}\cdot\sqrt[4]{b}. Notice that a^{4} is already a perfect 4^{\text{th}} power, so we do not need to write out its factors.

Apply the idea

\displaystyle -a\sqrt[4]{80a^{4}}	\displaystyle =	\displaystyle -a\sqrt[4]{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot a^{4}}	Find prime factorization of 80
	\displaystyle =	\displaystyle -a \sqrt[4]{2^{4} \cdot a^{4} \cdot 5}	Rewrite using perfect 4th powers
	\displaystyle =	\displaystyle -a \cdot \sqrt[4]{2^{4}} \cdot \sqrt[4]{a^{4}} \cdot \sqrt[4]{5}	Product of radicals property
	\displaystyle =	\displaystyle -a \cdot 2 \cdot a \cdot \sqrt[4]{5}	4th root of a perfect 4th power
	\displaystyle =	\displaystyle -2a^{2} \sqrt[4]{5}	Simplify

Example 3

Assume that x is non-negative, simplify 3x\sqrt{-81x^{5}}.

Worked Solution

Create a strategy

To simplify this expression, we'll first break down \sqrt{-81x^{5}} into its perfect square factors and separate the square root of a negative number using i, the imaginary unit.

Apply the idea

\displaystyle 3x\sqrt{-81x^{5}}	\displaystyle =	\displaystyle 3x \cdot \sqrt{9^{2} \cdot -1 \cdot x^{4} \cdot x }	Rewrite using perfect squares
	\displaystyle =	\displaystyle 3x \cdot \sqrt{9^{2}} \cdot \sqrt{-1} \cdot \sqrt{x^{4}} \cdot \sqrt{x}	Product of radicals property
	\displaystyle =	\displaystyle 3x \cdot 9 \cdot \sqrt{-1} \cdot x^{2} \cdot \sqrt{x}	Square root of perfect square
	\displaystyle =	\displaystyle 3x \cdot 9 \cdot i \cdot x^{2} \cdot \sqrt{x}	Definition of i
	\displaystyle =	\displaystyle 27x^{3}i \sqrt{x}	Evaluate the multiplication

Reflect and check

Notice that if x had represented a negative number, the radicand would have been a positive number and the simplified expression would not be imaginary. However, the instructions said x represents a non-negative value, so the result cannot be real.

Example 4

Simplify the algebraic radical: x^{3}y\sqrt[3]{x^{6}y^{15}}.

Worked Solution

Create a strategy

First, we can rewrite the factors in the radicand as powers of 3. Then, we can remove the powers of 3 from under the radical and multiply the results to the variables already outside the radical.

Apply the idea

\displaystyle x^{3}y\sqrt[3]{x^{6}y^{15}}	\displaystyle =	\displaystyle x^{3}y\sqrt[3]{\left(x^{2}\right)^{3} \cdot \left(y^{5}\right)^{3}}	Power rule
	\displaystyle =	\displaystyle x^3y \cdot \sqrt[3]{(x^{2})^{3}}\cdot \sqrt[3]{(y^5)^{3}}	Product of radicals property
	\displaystyle =	\displaystyle x^{3}y \cdot x^{2} \cdot y^{5}	Evaluate cube roots
	\displaystyle =	\displaystyle x^{5}y^{6}	Product rule

Idea summary

Radical expressions can be simplified by grouping factors and simplifying a product of radicals.

\displaystyle \sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}

\bm{a,b}

are positive integers or variables

\bm{n}

is a positive integer

Simplifying also uses the property of taking the n^{th} root of an n^{th} power

\displaystyle \sqrt[n]{a^{n}}=a

\bm{a}

is a positive integer or variable

\bm{n}

is a positive integer

The power of powers property groups factors.

\displaystyle \left(a^{m}\right)^{n}=a^{mn}

\bm{a}

is a positive integer or variable

\bm{m,n}

is a positive integer

When the radicand is negative in a square root function, the simplified radical will result in an imaginary number, i.

Outcomes

A2.EO.2

The student will perform operations on and simplify radical expressions.

A2.EO.2a

Simplify and determine equivalent radical expressions that include numeric and algebraic radicands

4.01 Simplify radicals

Ideas

Simplifying radical expressions

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

A2.EO.2

A2.EO.2a

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