This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Simply Math
- Business MathToggle Dropdown
- Hospitality MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering MathToggle Dropdown
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-Health
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Nutrition Labels
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- PhysicsToggle Dropdown

There are three rules a radical expression must obey in order to be considered fully simplified.

- No square numbers as factors under the root sign.
- No fractions under the root sign.
- No square roots in the denominator.

Note: A square number is one that can be obtained by taking the square of a smaller number. For example, 4, 9, 16, 25, 36, and 49 are square numbers because \( \qquad 2^2 = 4 ,\qquad 3^2 = 9 ,\qquad 4^2 = 16 ,\qquad \) and so on...

`Example`

The following radicals are NOT fully simplified.

\[ \sqrt{18}, \quad \sqrt{\frac{1}{2}}, \quad \frac{3}{\sqrt{5}} \]

- \(\sqrt{18}\) because \(9\) is a factor of \(18\) and \(9\) is a square number (\(3^2\)).
- \( \sqrt{\frac{1}{2}} \) because there is a fraction underneath the square root.
- \( \frac{3}{\sqrt{5}} \) because the \(\sqrt5\) is in the denominator.

We will discuss how to simplify these types of radicals later on in the lesson.

To remove square factors from below square root signs, try to find the largest square factor of the number and factor the number accordingly.

`Example`

Let's simplify \(\sqrt{200}\).

`Solution`

Notice that \(200 = 100 \times 2\), so

\[ \begin{align*} \sqrt{200} &= \sqrt{100 \times 2} \\ &= \sqrt{100}\times\sqrt{2} \\ &= 10\sqrt{2} \end{align*} \]

But what if you don't notice that \(100\) is the largest square factor? If you have trouble identifying the largest square factor, you can always use the prime factorisation of the number to reduce. See the section on Prime Factorisation and Least Common Multiple for more details on this.

First, find the prime factorisation of 200.

\[ \begin{align*} 200 &= 100 \times 2 \\ &= 50 \times 2 \times 2 \\ &= 25 \times 2 \times 2 \times 2 \\ &= 5 \times 5 \times 2 \times 2 \times 2 \end{align*} \]

Therefore,

\[ \begin{align*} \sqrt{200} &= \sqrt{5 \times 5 \times 2 \times 2 \times 2} \\ &= \sqrt{5} \times \sqrt{5} \times \sqrt{2} \times \sqrt{2} \times \sqrt{2} \end{align*} \]

Now, notice that

\[ \sqrt{5} \times \sqrt{5} = (\sqrt{5})^2 = 5 \qquad \text{ and likewise,} \qquad \sqrt{2} \times \sqrt{2} = 2 \]

so

\[ \begin{align*} \sqrt{200} &= \sqrt{5} \times \sqrt{5} \times \sqrt{2} \times \sqrt{2} \times \sqrt{2} \\ &= 5 \times 2 \times \sqrt2 \\ &= 10\sqrt2 \end{align*} \]

Once the number is written out as a product of square roots of prime numbers, the numbers can be __paired up and reduced__ as shown in the example above.

`Example`

Suppose you find \(\sqrt{3}\times\sqrt{3}\times\sqrt{3}\times\sqrt{3}\times\sqrt{3}\) in the reduction. There are five \(\sqrt3\)'s in this expression, so group them as \((\sqrt{3}\times\sqrt{3})\times(\sqrt{3}\times\sqrt{3})\times\sqrt{3}\) and reduce as above to get \(9\sqrt3\)

Fractions under roots can be be dealt with using the following property:

\[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

`Examples`

- \( \sqrt{\dfrac{3}{4}} = \dfrac{\sqrt{3}}{\sqrt{4}} = \dfrac{\sqrt{3}}{2} \)
- \( \sqrt{\dfrac{7}{3}} = \dfrac{\sqrt{7}}{\sqrt{3}} \)

Notice that removing the square root from below the root sometimes leads to a square root below the denominator as in the second example above. This can be fixed by multiplying above and below by the root term in the denominator.

`Example`

Using the above example, we ended up at:

\[ \sqrt{\frac{7}{3}} = \frac{\sqrt{7}}{\sqrt{3}} \]

with a \(\sqrt3 \) in the denominator. To remove it, we multiply above and below by \(\sqrt3\).

\begin{align} \frac{\sqrt{7}}{\sqrt{3}} &= \frac{\sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ &= \frac{\sqrt{7}\sqrt3}{3} \\ &= \frac{\sqrt{21}}{3} \end{align}

`Example`

Simplify the radical expression \(\sqrt{\dfrac{3}{160}}\)

`Solution`

- Apply the radical to both numerator and denominator. \[ \sqrt{\dfrac{3}{160}} = \dfrac{\sqrt3}{\sqrt{160}} \]
- Reduce \(\sqrt{160}\). \[ \sqrt{160} = \sqrt{16}\sqrt{10} = 4\sqrt{10} \]
- Remove the \(\sqrt{10}\) from the denominator.

\begin{align} \sqrt{\dfrac{3}{160}} &= \dfrac{3}{4\sqrt{10}} \\ &= \dfrac{\sqrt3}{4\sqrt{10}} \times \dfrac{\sqrt{10}}{\sqrt{10}} \\ &= \frac{\sqrt{30}}{40} \end{align}

Now the expression satisfies all the rules to be considered fully simplified.

- Last Updated: Mar 4, 2024 10:18 AM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
- Print Page

chat loading...