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Wize Grade 11 Mathematics Textbook > Radical Expressions and Functions (Content Coming Soon)

Operations with Radicals

Graphing Polynomials using TransformationsPrevious Section
Solving Radical EquationsNext Section

Radical Expressions

The square root is one basic type of radical expression. In a radical expression we have
  • radical symbol
  • radicand
  • index



The value of a radical can be found be finding the expression, which when raised to the power of the index, will equal the radicand.



For even roots, we take this value to be positive and call it the principle root of the radial.

Example 1

1. 49\sqrt{49}

49=7⋅7=7\begin{aligned} \sqrt{49} &= \sqrt{7 \cdot 7} \\ &= 7 \end{aligned}


2. 643\sqrt[3]{64}

643=4⋅4⋅43=4\begin{aligned} \sqrt[3]{64} &= \sqrt[3]{4 \cdot 4 \cdot 4} \\ &= 4 \end{aligned}


3. x4y2\sqrt{x^4y^2}

x4y2=(x2y)⋅(x2y)=x2y\begin{aligned} \sqrt{x^4y^2} &= \sqrt{(x^2y) \cdot (x^2y)} \\ &= x^2y \end{aligned}

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Alternate Notation

All radicals can be re-written using exponents. To write a radical as an exponent any powers on the radical are used as the numerator, and the index of the root is used as the denominator.



Example 2

1. xy\sqrt{xy}

xy=(xy)1/2\begin{aligned} \sqrt{xy} &= (xy)^{1/2} \\ \end{aligned}

2. x45\sqrt[5]{x^4}

x45=x4/5\begin{aligned} \sqrt[5]{x^4} &= x^{4/5} \\ \end{aligned}

3. (y)3(\sqrt{y})^3

(y)3=y3/2\begin{aligned} (\sqrt{y})^3 &= y^{3/2} \\ \end{aligned}


Wize Tip
Somethings are easier to recognize using the alternate notation. Try to get comfortable being able to switch between the two different notations.


Operations with radicals

We can combine radicals in a variety of ways according to their rules



Example 1
Combine the following expressions

1. xy2⋅x2y3\sqrt{xy^2} \cdot\sqrt{x^2y^3}

xy2⋅x2y3=(xy2)(x2y3)=x3y5\begin{aligned} \sqrt{xy^2}\cdot\sqrt{x^2y^3} &= \sqrt{(xy^2)(x^2y^3)} \\ &= \sqrt{x^3y^5} \end{aligned}

2. 2x+4x2\sqrt{x} + 4\sqrt{x}

2x+4x=(2+4)x=6x\begin{aligned} 2\sqrt{x} + 4\sqrt{x} &= (2+4)\sqrt{x} \\ &= 6\sqrt{x} \end{aligned}

3. 2q3⋅3z\sqrt[3]{2q} \cdot \sqrt{3z}

2q3⋅3z=(2q)1/3⋅(3z)1/2=(2q)2/6⋅(3z)3/6=((2q)2(3z)3)1/6=(2q)2(3z)36=4q2⋅27z36=108q2z36\begin{aligned} \sqrt[3]{2q} \cdot \sqrt{3z} &= (2q)^{1/3} \cdot (3z)^{1/2} \\ &= (2q)^{2/6} \cdot (3z)^{3/6} \\ &= ((2q)^2(3z)^3 )^{1/6} \\ &= \sqrt[6]{(2q)^2(3z)^3} \\ &= \sqrt[6]{4q^2\cdot27z^3} \\ &= \sqrt[6]{108q^2z^3} \end{aligned}



Wize Tip
The rules can be described using the alternate notation for radicals. When this is done we can see that these are actually the same rules for exponents.




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Simplifying Radicals

When simplifying a radical expression we want to make sure that:
  • The powers in the radicand are smaller than the index
  • The powers in the radicand and the index do not have a common factor more than 1
  • The powers in the radicand are not fractions
  • A radical does not appear in any denominators
The process of making sure a radical is not in the denominator is called rationalizing the denominator.

Example 2
Simplify the following radical expression

1. y(x+2)4\sqrt{y(x+2)^4}

y(x+2)4=(x+2)2y\begin{aligned} \sqrt{y(x+2)^4} &= (x+2)^2\sqrt{y} \end{aligned}

2. x(x+1)3\displaystyle\frac{x}{\sqrt[3]{(x+1)}}

x(x+1)3=x(x+1)23(x+1)3(x+1)23=x(x+1)23(x+1)33=x(x+1)23x+1\begin{aligned} \displaystyle\frac{x}{\sqrt[3]{(x+1)}} &= \displaystyle\frac{x\sqrt[3]{(x+1)^2}}{\sqrt[3]{(x+1)}\sqrt[3]{(x+1)^2}} \\ &= \frac{x\sqrt[3]{(x+1)^2}}{\sqrt[3]{(x+1)^3}} \\ &= \frac{x\sqrt[3]{(x+1)^2}}{x+1} \end{aligned}

Example: Operations with Radicals


At an assembly factory information kept on how long it takes to build machinery depending on the amount of design changes.
This information is then used to build prediction models.
For example the approximate number of months required to assemble machine A and machine B is given by

A(x)=(x+3)23A(x) = \sqrt[3]{(x + 3)^2}

B(x)=(x+3)53B(x) = \sqrt[3]{(x+3)^5}

where xx is the number of design changes.

Create a new model that represents the total number of months for assembling both machines A and B.
Simplify as much as possible.

For this we would be begin by adding the two current models together.

C(x)=A(x)+B(x)=(x+3)23+(x+3)53\begin{aligned} C(x) & = A(x) + B(x) \\ &=\sqrt[3]{(x+3)^2} + \sqrt[3]{(x+3)^5} \\ \end{aligned}

Now we can simplify one of the roots since the power is larger than the index

C(x)=(x+3)23+(x+3)53=(x+3)23+(x+3)3(x+3)23=(x+3)23+(x+3)(x+3)23\begin{aligned} C(x) &=\sqrt[3]{(x+3)^2} + \sqrt[3]{(x+3)^5} \\ &=\sqrt[3]{(x+3)^2} + \sqrt[3]{(x+3)^3(x+3)^2} \\ &=\sqrt[3]{(x+3)^2} + (x+3)\sqrt[3]{(x+3)^2} \\ \end{aligned}

Since the radicands are the same, we can add by combining their coefficients

C(x)=(x+3)23+(x+3)(x+3)23=(1+x+3)(x+3)23=(x+4)(x+3)23\begin{aligned} C(x) & = \sqrt[3]{(x+3)^2} + (x+3)\sqrt[3]{(x+3)^2} \\ &=(1 + x + 3)\sqrt[3]{(x+3)^2} \\ &=(x + 4)\sqrt[3]{(x + 3)^2} \end{aligned}
This gives us a new model that now represents the total time for making both machines.

Practice: Operations with Radicals

Simplify the following radical expressions:
(2x+1)4\sqrt{(2x + 1)^4}

Practice: Operations with Radicals

Combine the radical expressions and then simplify

1. 327y−75y+212y3\sqrt{27y} - \sqrt{75y} + 2\sqrt{12y}

2. (2xy+1)(xy−3)(2\sqrt{xy} + 1)(\sqrt{xy} - 3)

3. 2p6+7p12\sqrt{\displaystyle\frac{2}{p^6}} + \sqrt{\displaystyle\frac{7}{p^{12}}}

Practice: Operations with Radicals

The expressions below have radicals in the denominator.
What must be multiplied in the numerator and denominator to remove the radical in the denominator?

1. 5y\displaystyle\frac{5}{\sqrt{y}}


2. y+2y−3\displaystyle\frac{y+2}{\sqrt{y} -3}


3. 3x3+x−y\displaystyle\frac{3x}{3 + \sqrt{x - y}}