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5. Rational Exponents & Radical Functions
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United States of AmericaFL
Grade 912

5.02 Operations with radicals

Lesson
Practice
Lesson

Concept summary

The same operations that apply to numeric radicals can also be applied to algebraic 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.a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}, \text{ for }x,y \geq 0
  • 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.\dfrac{a\sqrt[n]{x}}{b\sqrt[n]{y}}=\dfrac{a}{b}\sqrt[n]{\dfrac{x}{y}}, \text{ for }x,y \geq 0
  • 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.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}

Worked examples

Example 1

Assuming that each variable represents a positive number, fully simplify each of the following expressions, writing them as a single radical:

a

\sqrt[3]{125y}+2\sqrt[3]{y}

Approach

As these are not like radicals we cannot add them together as is. We want to check if we can simplify one or both such that they can be simplified.

125 is a perfect cube, and so we can simplify \sqrt[3]{125y} and check if we now have like radicals that can be added together.

Solution

\displaystyle \sqrt[3]{125y}+2\sqrt[3]{y}\displaystyle =\displaystyle \sqrt[3]{5^3y}+2\sqrt[3]{y}Express 125 with an exponent of 3
\displaystyle =\displaystyle \sqrt[3]{5^3}\sqrt[3]{y}+2\sqrt[3]{y}Use \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}
\displaystyle =\displaystyle 5\sqrt[3]{y}+2\sqrt[3]{y}Evaluate the numeric radical
\displaystyle =\displaystyle 7\sqrt[3]{y}Combine like radicals
b

3\sqrt[3]{4x^2}\cdot 5\sqrt[3]{16x}

Approach

We want to simplify the product of two algebraic radicals with numeric factors in the radicand and also numeric coefficients. We can approach this in a couple of different ways:

  • Separating the numeric and algebraic products and then combining like with like
  • Combining the radicals as they are presented and then separating the numeric and algebraic products

Solution

\displaystyle 3\sqrt[3]{4x^2}\cdot 5\sqrt[3]{16x}\displaystyle =\displaystyle 15\sqrt[3]{4\left(16\right)xx^2}Use a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}
\displaystyle =\displaystyle 15\sqrt[3]{64x^3}Simplify the products
\displaystyle =\displaystyle 15\sqrt[3]{4^3x^3}Express 64 as a power of 3
\displaystyle =\displaystyle 15\sqrt[3]{4^3}\sqrt[3]{x^3}Use \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}
\displaystyle =\displaystyle 60xEvaluate the radicals

Reflection

If we chose to separate the numeric and algebraic products in the radicand first, our work to find the solution would be similar to the following:

\displaystyle 3\sqrt[3]{4x^2}\cdot 5\sqrt[3]{16x}\displaystyle =\displaystyle 15\sqrt[3]{4}\sqrt[3]{16}\sqrt[3]{x^2}\sqrt[3]{x}
\displaystyle =\displaystyle 15\sqrt[3]{64}\sqrt[3]{x^3}
\displaystyle =\displaystyle 15\left(4\right)x
\displaystyle =\displaystyle 60x
c

\dfrac{\sqrt{20p^3}}{\sqrt{125p^2}}

Approach

Similar to part (b) we can approach this expression in a few ways, either combining and then simplifying, or simplifying then combining.

Notice that, the numeric individual radicands of 20 and 125 are not perfect squares, but \\ \dfrac{20}{125}=\dfrac{4}{25} which has perfect squares in the numerator and the denominator, so we will first write the expression as a single radical.

Solution

\displaystyle \dfrac{\sqrt{20p^3}}{\sqrt{125p^2}}\displaystyle =\displaystyle \sqrt{\frac{20p^3}{125p^2}}Use \dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}=\sqrt[n]{\dfrac{ax}{by}}
\displaystyle =\displaystyle \sqrt{\frac{4p}{25}}Simplify the quotient
\displaystyle =\displaystyle \frac{\sqrt{4p}}{\sqrt{25}}Use \sqrt[n]{\dfrac{ax}{by}}=\dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}
\displaystyle =\displaystyle \frac{2\sqrt{p}}{5}Simplify the radical

Reflection

If we chose to first simplify the radicals, our solution would like similar to the following:

\displaystyle \dfrac{\sqrt{20p^3}}{\sqrt{125p^2}}\displaystyle =\displaystyle \frac{\sqrt{5\cdot 4 p^2 p}}{\sqrt{5\cdot 25 p^2}}
\displaystyle =\displaystyle \frac{2p\sqrt{5p}}{5p\sqrt{5}}
\displaystyle =\displaystyle \frac{2\sqrt{5p}}{5\sqrt{5}}
\displaystyle =\displaystyle \frac{2\sqrt{p}}{5}

Example 2

Fully simplify the following expression, where k \geq 0: \left(7 - 3\sqrt{k}\right)\left(2 + \sqrt{k}\right)

Approach

We can use the distributive property:\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd and then combine like terms to simplify.

Solution

\displaystyle \left(7 - 3\sqrt{k}\right)\left(2 + \sqrt{k}\right)\displaystyle =\displaystyle 7\left(2\right)+7\sqrt{k}-3\sqrt{k}\left(2\right)-3\sqrt{k}\sqrt{k}Distributive property
\displaystyle =\displaystyle 14+7\sqrt{k}-6\sqrt{k}-3kSimplify the products
\displaystyle =\displaystyle 14+\sqrt{k}-3kCombine like terms

Outcomes

MA.912.NSO.1.3

Generate equivalent algebraic expressions involving radicals or rational exponents using the properties of exponents.

MA.912.NSO.1.5

Add, subtract, multiply and divide algebraic expressions involving radicals.

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