You can expect to encounter some algebra questions while taking the PCAT. It’s important to know that these questions may be “disguised” as word problems. Make sure to read each question carefully and identify exactly what it is asking you.
Let’s review some algebraic concepts.
Exponents and Radicals
We often encounter exponents as numbers that are squared (such as 4²) or as square roots (such as √4). Remember, the square root is just the opposite of a squared number (√4 = 2). The √ symbol is called a radical.
You are probably also used to seeing cubed numbers (such as 4³) since these are used to describe volume. A cube root is the opposite of a cubed number (∛64 = 4).
Although squared and cubed numbers are probably most frequently encountered, remember that numbers may be raised to many different powers. Any number raised to the power of one is equal to itself and any number (with the exception of zero) raised to the power of zero equals one. Here are a few examples:
5¹ = 5
2,600¹ = 2,600
52⁰ = 1
8⁰ = 1
So, what if an exponent is a fraction?
An exponent of ½ is actually a square root:
An exponent of ⅓ is actually a cube root:
Remember that when multiplying two numbers with exponents, you must add the exponents. For example:
2³ ∙ 2² = 2⁵
When dividing a number with an exponent by another number with an exponent, you must use subtraction:
2³ ÷ 2² = 2¹ = 2
Let’s try an example equation that uses exponents and radicals.
First, let’s solve the multiplication in the first set of parentheses:
Remember that a number raised to the power of ½ is the square root of that number. The second set of parentheses encloses the square root of 25.
Next, we will raise 80 to the power of 0:
Then, we’ll find the square root of 25 again:
There are two 5s that do not have exponents. This means that they are each raised to the power of one:
We solved the equation! Next, we’ll review linear equations.
Linear equations are used to describe lines on coordinate planes. The standard form of a linear equation is ax + by = c.
Here is an example of a linear equation in standard form:
2x – 4y = 2
Not all linear equations are presented in standard form. Here are a few examples of linear equations that are not written in standard form:
y − 5 = −3(x + 5)
2x = -4y
3x – 4 + 4 = -10 + 4
x + 9 – 9 = 12 – 9
A system of linear equations refers to two or more equations that work together. Let’s look at an example of a system of linear equations and solve it:
y = 2x + 4
y = 3x + 2
Since each equation equals y, each equation is equivalent:
2x + 4 = 3x + 2
Move the x variables to one side by subtracting 2x from 3x. Remember that x by itself means 1x.
4 = x + 2
If 2 is subtracted from each side, we find that x = 2. Now you know that 2 is the value of x, so you can plug 2 into the second equation and solve it.
y = 3x + 2
y = 3(2) + 2
y = 6 + 2
y = 8
We find that in this system of linear equations, x = 2 and y = 8.
A quadratic equation describes a parabola when it is graphed. You can see an example of a parabola below.
Notice that (-3, 6) and (-1, 4) are two points on this particular parabola.
The standard form of a quadratic equation is ax² + bx + c = 0. The quadratic formula, shown below, is used to solve these equations.
Let’s look at an example of a quadratic equation and solve it:
6x² − 5x − 3=0
Here, 6 = a, -5 = b, and -3 = c. We can plug the values into the quadratic equation formula:
Notice that the 5 outside of the radical is negative; in other words, when multiplied it becomes a positive. Let’s go ahead and square the -5 inside the radical.
Next, we will perform the multiplication:
Notice that there are really two answers here because of the plus or minus (±) sign. Remember that a parabola is curved and what appears on one side is mirrored on the other side. That’s why two “opposite” answers are correct: they mirror one another.