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4. Radical Functions
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Algebra 2

4.03 Add and subtract radical expressions

Lesson
Practice

Add and subtract radical expressions

The same operations that apply to numeric radicals can also be applied to algebraic radical expressions.

We can add or subtract like radicals (radicals with the same index and radicand) by adding the coefficients and keeping the radicand the same.

a\sqrt[n]{x}+b\sqrt[n]{x}=\left(a+b\right)\sqrt[n]{x}

\text{or}

a\sqrt[n]{x}-b\sqrt[n]{x}=\left(a-b\right)\sqrt[n]{x}

If there are no like radicals, check to see if any of the radicals can be simplified first.

Examples

Example 1

Assuming y>0, simplify 10\sqrt{2y}+14\sqrt{2y}

Worked Solution
Create a strategy

Because the radicals have the same radicand, we can add the coefficients and keep the radicand the same.

Apply the idea
\displaystyle 10\sqrt{2y}+14\sqrt{2y}\displaystyle =\displaystyle 24\sqrt{2y}Add the coefficients
Reflect and check

In the provided simplification, we assume that y \gt 0 to ensure the expression under the square root is positive, which allows us to directly add the coefficients. However, if y were negative, say y=-2, the expression inside the square root becomes negative, leading us to consider complex numbers.

\displaystyle 10\sqrt{2y}+14\sqrt{2y}\displaystyle =\displaystyle 10\sqrt{2(-2)}+14\sqrt{2(-2)}Substitute y=-2
\displaystyle =\displaystyle 10\sqrt{-4}+14\sqrt{-4}Evaluate the multiplication
\displaystyle =\displaystyle 10i\sqrt{4}+14i\sqrt{4}Rewrite negative radicands with i, the imaginary unit
\displaystyle =\displaystyle 24i\sqrt{4}Add the coefficients
\displaystyle =\displaystyle 24i(2)Simplify the square root
\displaystyle =\displaystyle 48iEvaluate the multiplication

Hence, if y=-2, simplifying the original expression would result in 48i, introducing a factor of i due to the square root of a negative number.

Example 2

Simplify \sqrt[3]{125y}+\sqrt[3]{8y}

Worked Solution
Create a strategy

Because the radicals are not alike, we cannot combine them. Instead, notice that the radicands contain perfect cube factors. First, we will simplify each term, then add the coefficients of the simplified, like radicals.

Apply the idea
\displaystyle \sqrt[3]{125y}+\sqrt[3]{8y}\displaystyle =\displaystyle \sqrt[3]{5^3\cdot y}+\sqrt[3]{2^3\cdot y}Express 125 and 8 with exponents of 3
\displaystyle =\displaystyle 5\sqrt[3]{y}+2\sqrt[3]{y}Simplify each cube root
\displaystyle =\displaystyle 7\sqrt[3]{y}Combine like terms
Reflect and check

Remember that \sqrt[3]{a} + \sqrt[3]{b} \neq \sqrt[3]{a+b}. This means you cannot add radicals if the radicands are different.

For example, if we had attempted to combine the radicands first, the expression would have incorrectly simplified to \sqrt[3]{133y}, which does not lead to the same answer as what we found.

Anytime the radical terms in an sum or difference are not alike, always check to see if the radicals can be simplified first. After simplifying each radical, check that the resulting radicands are the same before combining the terms.

Example 3

Simplify \sqrt[3]{512v}-5\sqrt[3]{v}

Worked Solution
Create a strategy

Since the radicands are different, simplify each term, then subtract the like radicals.

Apply the idea
\displaystyle \sqrt[3]{512v}-5\sqrt[3]{v}\displaystyle =\displaystyle \sqrt[3]{512}\sqrt[3]{v}-5\sqrt[3]{v}Write the radicals as a products of their factors
\displaystyle =\displaystyle 8\sqrt[3]{v}-5\sqrt[3]{v}Simplify the cube root using 8^3=512
\displaystyle =\displaystyle 3\sqrt[3]{v}Subtract the like radicals
Reflect and check

Notice the expression 3\sqrt[3]{v} is fully simplified because there are no perfect cube factors in the radicand, and no other algebraic operations need to be performed.

Example 4

Simplify 6\sqrt[5]{7a}+7\sqrt[5]{5a}-3\sqrt[5]{7a}+8\sqrt[5]{5a}

Worked Solution
Create a strategy

Group the like radicals and then add the coefficients of like radicals.

Apply the idea
\displaystyle 6\sqrt[5]{7a}+7\sqrt[5]{5a}-3\sqrt[5]{7a}+8\sqrt[5]{5a}\displaystyle =\displaystyle (6\sqrt[5]{7a}-3\sqrt[5]{7a})+(7\sqrt[5]{5a}+8\sqrt[5]{5a})Group like radicals
\displaystyle =\displaystyle 3\sqrt[5]{7a}+15\sqrt[5]{5a}Simplify like radicals
Reflect and check

Simplifying this expression is similar to simplifying the algebraic expression 6x+7y-3x+8y. We identify the like terms, then add and subtract the coefficients of similar terms.

Similar to how 6x-3x becomes 3x, we subtracted the like terms 6\sqrt[5]{7a}-3\sqrt[5]{7a} to get 3\sqrt[5]{7a}. And similar to how 7y+8y simplifies to 15y, 7\sqrt[5]{5a}+8\sqrt[5]{5a} simplifies to 15\sqrt[5]{5a}.

Idea summary

When adding and subtracting algebraic radicals, they must have the same radicand before we can add or subtract like radicals.

a\sqrt[n]{x}+b\sqrt[n]{x}=\left(a+b\right)\sqrt[n]{x}

\text{or}

a\sqrt[n]{x}-b\sqrt[n]{x}=\left(a-b\right)\sqrt[n]{x}

Outcomes

A2.EO.2

The student will perform operations on and simplify radical expressions.

A2.EO.2b

Add, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator.

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