Rules and operations

Absolute value

Definitions

Absolute value

The distance a number is from zero

Absolute value is the distance a number is from zero. Because a distance cannot be negative, the end result of an absolute value expression is always positive.

$$\begin{array}{rl}|4|& =4\\ |-4|& =4\end{array}$$

When working with equations that contain absolute value, there will be two possible answers: a positive and a negative one. Each value will make the equation true because your end result always turns positive.

Example:

$$|-6x|=30$$

To solve this equation, we need to get $x$ by itself. $-6x$ could equal either positive $30$ or negative $30$, as the absolute value sign would turn it positive. So we need to create two equations and set $-6x$ to both $30$ and $-30$ to find both possible values for $x$ that would make the equation true.

$$\begin{array}{rl}-6x& =30\\ \frac{\cancel{-6}x}{\cancel{-6}}& =\frac{30}{-6}\\ x& =-5\end{array}$$

$$\begin{array}{rl}-6x& =-30\\ \frac{\cancel{-6}x}{\cancel{-6}}& =\frac{-30}{-6}\\ x& =5\end{array}$$

So our solutions are $x=5$ and $x=-5$!

Steps taken:

• Split into two cases
• Solve each case

Try one on your own!

Knowledge check: Solve $|-5+5x|=20$?

(spoiler)

$x=5$ and $x=-3$

Proportions

Definitions

Proportion

The relationship between two quantities that compares two equal ratios. Proportions are useful for finding an unknown value that keeps the ratios equivalent.

A proportion shows that two fractions (ratios) are equal. You’ll often use proportions to solve for a missing value.

For example:

$$\frac{1}{2}=\frac{x}{10}$$

To solve, you are looking for the value of $x$ that makes both sides equal.

One common way to solve proportions is cross multiplication (sometimes called the butterfly method). To use it:

• Multiply the diagonals of the two fractions
• Set those products equal to each other
• Solve for the unknown This works because if two fractions are equal, the product of the cross terms will also be equal. Let’s practice!

Example 1:

$$\begin{array}{rl}\frac{2}{5}& =\frac{x}{10}\\ 2\cdot 10& =5x\\ 20& =5x\\ \frac{20}{5}& =\frac{\cancel{5}x}{\cancel{5}}\\ 4& =x\end{array}$$

Example 2:

$$\begin{array}{rl}\frac{5}{m-2}& =\frac{6}{18}\\ 5\cdot 18& =(m-2)\cdot 6\\ 90& =6m-12\\ 90+12& =6m-\cancel{12}\cancel{(+12)}\\ 102& =6m\\ \frac{102}{6}& =\frac{\cancel{6}m}{\cancel{6}}\\ 17& =m\end{array}$$

Exponents

Definitions

Exponents

The power that represents how many times a base number is being multiplied by itself

Exponents represent repeated multiplication of a base. For example, $3^4=3\cdot 3\cdot 3\cdot 3$.

When working with exponents, there are a few key rules. You don’t need to memorize the rule names, but you do need to know what each rule does.

Here are the main exponent patterns you’ll use:

Name	Rule	Example
Zero Exponent Rule	$a^0=1$	$2^0=1$
Product Rule	$a^b\cdot a^c=a^{b+c}$	$2^3\cdot 2^4=2^7$
Quotient Rule	$\frac{a^b}{a^c}=a^{b-c}$	$\frac{2^4}{2^3}=2^1$
Power of Power Rule	$(a^b{)}^c=a^{b\cdot c}$	$(2^4{)}^3=2^{12}$
Power of Product Rule	$(ab{)}^c=a^c{b}^c$	$(2b{)}^3=2^3{b}^3$
Negative Exponent Rule	$a^{-b}=\frac{1}{a^b}$	$2^{-3}=\frac{1}{2^3}$

Example 1:

$$\begin{array}{rl}(4^2{)}^0& =4^{2\cdot 0}\\ & =4^0\\ & =1\end{array}$$

Example 2:

$$\begin{array}{rl}(x^2{y}^{-2}{)}^4& =x^{2\cdot 4}{y}^{-2\cdot 4}\\ & =x^8{y}^{-8}\\ & =\frac{x^8}{y^8}\end{array}$$

Steps taken:

• Apply exponent rules
• Simplify

Roots

Definitions

Roots

The reverse of an exponent, also called a radical, and can be written as fractional exponents (for example, $\sqrt{a}=a^{\frac{1}{2}}$)

Roots are the reverse of exponents. They help you find what number was multiplied to get a result.

Here’s how to convert from radical to exponential form:

$$\sqrt[c]{a^b}=a^{\frac{b}{c}}$$

The most common roots you’ll see on the ASVAB are square roots and cube roots. Knowing common squares and cubes will make these problems much faster. Many students memorize:

• Square numbers up to $12$
• Cube numbers up to $5$

Square roots to memorize

$\sqrt{1}=1$
$\sqrt{4}=2$
$\sqrt{9}=3$
$\sqrt{16}=4$
$\sqrt{25}=5$
$\sqrt{36}=6$
$\sqrt{49}=7$
$\sqrt{64}=8$
$\sqrt{81}=9$
$\sqrt{100}=10$
$\sqrt{121}=11$
$\sqrt{144}=12$

Cube roots to memorize

$\sqrt[3]{1}=1$
$\sqrt[3]{8}=2$
$\sqrt[3]{27}=3$
$\sqrt[3]{64}=4$
$\sqrt[3]{125}=5$

Example 1:

$$\begin{array}{rl}\sqrt{x^2}& =x^{2/2}\\ & =x\end{array}$$

Steps taken:

• Rewrite as an exponent
• Simplify

Simplifying radicals

When a problem includes roots, you’ll often be asked to simplify the radical as much as possible.

One common method is a factor tree. You start with the number inside the square root and break it into factors until you reach prime numbers. This is also called prime factorization.

Example 2:

Simplify:

$$\sqrt{63}$$

Break $63$ into $9\ast 7$.

>

$7$ is prime, so that branch stops. But $9$ can be broken into $3\ast 3$.

>

So the prime factorization of $63$ is $3\ast 3\ast 7$.

Because this is a square root, look for pairs of the same factor. The factor $3$ appears twice, so it forms a pair. You can pull one $3$ out of the square root, leaving $\sqrt{7}$ inside:

$\sqrt{63}=3\sqrt{7}$.

Steps taken:

• Factor into primes
• Find pairs
• Simplify

Sidenote

Multiple roots

If we were simplifying a cubic root, we would look for sets of $3$ identical factors to satisfy the root, for a fourth root, sets of $4$, etc. This is because roots undo exponents, so only complete groups can come out of the radical—any leftover factors stay inside.

Example 3:

$$\begin{array}{r}\sqrt{75}\\ \sqrt{25\cdot 3}\\ \sqrt{25}\cdot \sqrt{3}\\ 5\cdot \sqrt{3}\\ 5\sqrt{3}\end{array}$$

Steps taken:

• Factor the number
• Identify perfect squares (or full groups)
• Separate the root
• Simplify

Knowledge check 1: Convert to Radical Form $4^{\frac{2}{3}}$

(spoiler)

$\sqrt[3]{16}$ which can also be written as $\sqrt[3]{4^2}$

Knowledge check 2: Simplify $\sqrt{54}$

(spoiler)

$3\sqrt{6}$