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 a cube root \sqrt[3]{\quad}

Radicand

The value or expression underneath the radical symbol

Perfect square

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

Perfect cube

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

If a radical expression is written such that there are no factors that can be removed from the radicand, and with no radicals in the denominator (if the expression is a fraction), then the expression is said to be in simplified radical form.

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.

The same operations that can be applied to rational expressions can also be applied to radical expressions.

Multiplication: For radicals with the same index, multiply the coefficients, multiply the radicands, and write under a single radicand before checking to see if the radicand can be simplified further
Division: For radicals with the same index, divide the coefficients, divide the radicands, and write under a single radicand before checking to see if the radicand can be simplified further
Addition and subtraction: Add or subtract like radicals (radicals with the same index and radicand) by adding the coefficients and keeping the radicand the same, if there are no like radicals check to see if any of the radicals can be simplified first.

Worked examples

Example 1

Rewrite the expressions in simplified radical form.

\sqrt{150}

Approach

To simplify a square root it can be helpful to see if the radicand has any factors that are perfect squares. In this case, 150= 25 \cdot 6 and 25 is a perfect square.

Solution

Knowing this, we can simplify the original expression as follows:

\displaystyle \sqrt{150}	\displaystyle =	\displaystyle \sqrt{25\cdot6}
	\displaystyle =	\displaystyle \sqrt{25}\cdot\sqrt{6}
	\displaystyle =	\displaystyle 5\cdot\sqrt{6}

Since 6 does not have any factors that are perfect squares, the original expression is fully simplified as 5 \sqrt{6}.

\sqrt[3]{72}

Approach

To simplify a cube root it can be helpful to see if the radicand has any factors that are perfect squares. In this case, 72= 8 \cdot 9 and 8 is a perfect cube.

Solution

Knowing this, we can simplify the original expression as follows:

\displaystyle \sqrt[3]{72}	\displaystyle =	\displaystyle \sqrt[3]{8\cdot9}
	\displaystyle =	\displaystyle \sqrt[3]{8} \cdot \sqrt[3]{9}
	\displaystyle =	\displaystyle 2 \cdot \sqrt[3]{9}

Since 9 does not have any factors that are perfect cubes, the original expression is fully simplified as 2 \sqrt[3]{9}.

Example 2

Simplify the radical expressions.

\sqrt{8} \cdot \sqrt{5}

Approach

To simplify the product of two square roots we can multiply them together in the usual way. Then we can look at the result and factor out the square root of any perfect square factors, so that the solution is in simplified radical form.

Solution

Following this process, we have:

\displaystyle \sqrt{8} \cdot \sqrt{5}	\displaystyle =	\displaystyle \sqrt{8\cdot5}	Combining the radicals
	\displaystyle =	\displaystyle \sqrt{40}	Multiplying the radicands
	\displaystyle =	\displaystyle \sqrt{4\cdot 10}	Rewriting to find a perfect square factor
	\displaystyle =	\displaystyle 2\sqrt{ 10}	Taking out the perfect square factor

So the fully simplified form is 2 \sqrt{10}.

\frac{\sqrt[3]{48}}{\sqrt[3]{2}}

Approach

To simplify the quotient of two radicals it can be helpful to see if the radical in the numerator can be written as the product of two or more radicals, one of which is equal to the radical in the denominator.

Solution

In this case, 48 = 24 \cdot 2, and so we can rewrite the original expression as follows:

\displaystyle \frac{\sqrt[3]{48}}{\sqrt[3]{2}}	\displaystyle =	\displaystyle \frac{\sqrt[3]{24}\cdot\sqrt[3]{2}}{\sqrt[3]{2}}
	\displaystyle =	\displaystyle \sqrt[3]{24}
	\displaystyle =	\displaystyle \sqrt[3]{8 \cdot 3}
	\displaystyle =	\displaystyle 2\sqrt[3]{3}

So we have that \dfrac{\sqrt[3]{48}}{\sqrt[3]{2}} fully simplifies to 2 \sqrt[3]{3}.

Example 3

Simplify the following expressions:

9 \sqrt{3} + 11 \sqrt{3}

Approach

Since both terms in the expression include \sqrt{3} we can combine like terms to simplify.

Solution

9 \sqrt{3} + 11 \sqrt{3}=20 \sqrt{3}

4 \sqrt[3]{7} - 9 \sqrt[3]{7}

Approach

Since both terms in the expression include \sqrt[3]{7}, we can combine like terms to simplify.

Solution

4 \sqrt[3]{7} - 9 \sqrt[3]{7}=-5 \sqrt[3]{7}

\sqrt{180} + \sqrt{500}

Approach

Since the two terms in the expression have different radicands, we first want to rewrite each term in simplified radical form and see if they have any like terms. Since \sqrt{180}=6\sqrt{5} and \sqrt{500}=10 \sqrt{5} we can then combine like terms.

Solution

\displaystyle \sqrt{180} + \sqrt{500}	\displaystyle =	\displaystyle 6\sqrt{5} + 10 \sqrt{5}
	\displaystyle =	\displaystyle 16 \sqrt{5}

Outcomes

MA.912.NSO.1.1

Extend previous understanding of the Laws of Exponents to include rational exponents. Apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions involving rational exponents.

MA.912.NSO.1.2

Generate equivalent algebraic expressions using the properties of exponents.

MA.912.NSO.1.4

Apply previous understanding of operations with rational numbers to add, subtract, multiply and divide numerical radicals.

5.01 Operations with numerical radicals

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Example 2

Approach

Solution

Approach

Solution

Example 3

Approach

Solution

Approach

Solution

Approach

Solution

Outcomes

MA.912.NSO.1.1

MA.912.NSO.1.2

MA.912.NSO.1.4

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