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 or zero.

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

When working with equations that contain absolute value, there are usually two possible answers, because the expression inside the bars can equal either the positive or the negative of the number on the other side. Each answer makes the equation true because the absolute value turns both cases into the same positive number.

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)

$3\sqrt{6}$