Linear Function Definition

Before delving into radical functions, you should have a deep understanding of how functions work. We can best exemplify this with linear functions. A function takes a number as an input and gives a transformation of that number as an output.

 

function_rule
As you can see, the goal of functions is to show the relationship between the input and the output. This can be seen easily if we plot a function onto a graph.

 

linear_relationship_function
x f(x)
... ...
-1 1
0 4
1 7
... ...

 

What you can see in this graph is a linear function. Here is the standard way to write a linear function.

standard_form_function
f(x) Read as ‘f of x’ and can be thought of as y
m The slope
x The input
b The y-intercept

 

When we use functions, we use f of x, written as f(x), which can be thought of as what we usually call y. The linear function we used above can be written as:

linear_example

 

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Radicals

A radical function is any function that includes radicals. In algebra, radicals are also called roots. You may be familiar with how they look like:

radical_sign

This symbol is quite powerful - let’s take a look at a few examples of radicals to see why.

 

Name Example
\sqrt{}, \sqrt[^2]{} Square root, square \sqrt{4} = 2
\sqrt[^3]{} Cube root, cubic \sqrt[^3]{27} = 3
\sqrt[^x]{} Root \sqrt[^5]{32} = 5

A radical function, then, is simply a function that takes any number as an input and takes a root of it. Let’s take a basic example:

radical_function

The number inside of a square root, so we can plug in any number greater or equal to zero.

 

x f(x)
0 0.0
1 1.0
2 1.4
3 1.7
... ...

 

We can keep going, plotting each output up to 21. Next, we can graph the result.

exponential_equation

As you can see, the graph of a radical function is quite different from a linear function. Take a look at some more examples below.

cubic_relationship

cubic_function_example

exponential_function

 

Power Function

A power function is a function that involves a power. A power is also known as taking a number to the nth power. Take a look at the image below to get a better understanding.

exponential_growth_function

As you can see, we have a couple of elements involved here. The table below explains each element in the power function.

 

k Constant 3x^{2}, 4x, x, 10x
x Variable x,y,z,etc.
n Power 3x^{2},x^{4},2x^{3}

 

When you’re using power functions, you will probably come across two terms:

  • Growth function
  • Decay function

These two functions are found in many different disciplines and are usually used to model things with an exponential relationship. An exponential relationship is one modelled by an exponent, also known as a power. Take a look at a simple exponent relationship below.

exponential_growth

As you can see, ‘growth’ happens super quickly. While there’s not that much of a difference between 3 numbers, the difference between 9 to the power of 3 and 10 to the power of 3 is almost 300.

 

Radical and Power Rules

There are many rules when it comes to radicals and powers. First, let’s focus on radical rules. Take a look at the table below to get an idea of the rules and examples for each rule.

 

Rule Example
1 \sqrt^{2}{x} = \sqrt{x} \sqrt^{2}{4} = \sqrt{4}
2 \sqrt^{k}{x^{k}} = x \sqrt^{5}{20^{5}} = 20
3 \sqrt^{k}{x} * \sqrt^{k}{x} = \sqrt^{k}{x*x} \sqrt^{6}{5} * \sqrt^{6}{5} = \sqrt^{6}{25}

 

Now that you understand a bit more about radical rules, let’s take a look at some rules for powers.

 

Rule Example
1 x^{k} * x^{m} = x^{k+m} 4^{3} * 4^{2} = 4^{3+2}
2 \frac{x^{k}}{x^{m}} = x^{k-m} \frac{5^{6}}{5^{4}} = 5^{6-4}
3 (x^{k})^{m} = x^{k*m} (2^{3})^{2} = 2^{3*2}
4 (x^{-k}) = \frac{1}{x^{k}} (4^{-2}) = \frac{1}{4^{2}}
5 (x^{\frac{k}{m}}) = \sqrt^{m}{x^{k}} (8^{\frac{2}{3}}) = \sqrt^{3}{8^{2}}

 

Step-by-Step Factoring Radicals

Now that you’ve learned about radicals and how they relate to powers, let’s take a look at an example where we will factor a radical step-by-step. First, we will start with an easier example. The three examples after this one build upon each other.

 

Take a look at the radical below.

 

    \[ \sqrt^{3}{(2^{5})*(2^{-2})} \]

 

While this may look complex at first, we can start by focusing on what is inside of the fraction. The first step in simplifying this radical is to recall that negative exponents mean we have to take the inverse of the number.

 

    \[ \sqrt^{3}{(2^{5})*(\frac{1}{2^{2}})} \]

 

Next, since we are multiplying a whole number by a fraction, we can rewrite that into one simple fraction.

 

    \[ \sqrt^{3}{ \frac{ (2^{5}) }{ (2^{2}) } } \]

 

Now, we can see that since we have the same main number in the numerator and the denominator, we can use the power rules to simplify this fraction.

 

    \[ \sqrt^{3}{ 2^{5-2} } = \sqrt^{3}{ 2^{3} } = 2 \]

 

Since the nth square root of a number to the nth power is just the number, we can stop here and say the answer is 2.

 

Example 1

In this example, try to simplify the radical below by yourself first. Then, compare your solution with the following.

cubic_root

 

Take a look at the step-by-step solution below.

 

    \[ \sqrt^{3}{2^{4} * \frac{1}{\sqrt^{3}{8}}} \]

    \[ \sqrt^{3}{2^{4} *\frac{1}{2}} \]

    \[ \sqrt^{3}{\frac{2^{4}}{2}} \]

    \[ \sqrt^{3}{2^{4-1}} = \sqrt^{3}{2^{3}} = 2 \]

 

Example 2

In this example, try to simplify the radical below by yourself first. Then, compare your solution with the following.

cubic_root_example

Take a look at the step-by-step solution below.

 

    \[ \sqrt^{3}{250} \]

    \[ \sqrt^{3}{125*2} \]

    \[ \sqrt^{3}{125} * \sqrt^{3}{2} \]

    \[ 5(\sqrt^{3}{2}) \]

 

Example 3

In this example, try to simplify the radical below by yourself first. Then, compare your solution with the following.

square_root_example

Take a look at the step-by-step solution below.

 

    \[ (\sqrt{74})^{6} \]

    \[ (74^{1/2})^{6} \]

    \[ 74^{1/2*6} = 74^{3} \]

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.