In this article, we’ll learn all about exponents and radicals. We’ll also get a chance to practice these concepts on problems similar to those you might see on the GED Math Section.
(For more information about what’s on the test, check out: What’s on the GED Math Test?)
What are Exponents?
Let’s start with the basics. Whole number Exponents, or powers, are shorthand for repeated multiplication.
For example, 43 = 4 × 4 × 4 = 16 × 4 = 64. (Be careful: This is not the same thing as 4 × 3, which equals 12.)
Exponents make it quicker and more efficient to write algebraic expressions involving lots of factors of the same kind. So, instead of writing an expression like this…
3 × 3 × 3 × 3 × x × x × x × x × x × y × y
… you would write it instead like this:
Of course this works the other way around too. If you need to evaluate (-2)3, for example, then first write it as repeated multiplication: (-2)3 = (-2)(-2)(-2). Then work through the products to get (-2)(-2)(-2) = (4)(-2) = -8. (This is really important for the no-calculator section of the test!)
What if the exponent is 1? Well think about it… if there’s only one factor, then there’s nothing else to multiply to it. So x1 = x for any number x.
We’ll talk about other kinds of exponents below, including zero, negative, and fractional.
Order of Operations
According to standard order of operations, evaluating exponents comes before any multiplications, divisions, additions or subtractions. Keep in mind that grouping symbols always take precedence though.
Just remember “PEMDAS” (Please Excuse My Dear Aunt Sally).
- P — Parentheses and other grouping symbols must be worked out as a self-contained group first.
- E — Exponents come next.
- M and D — Multiplication and Division occur next, going left to right.
- A and S — Addition and Subtraction are done last, again going from left to right.
- 6 – 23 = 6 – 8 = -2. (NOT 6 – 23 = 43 = 64.)
- 5 · 22 = 5 · 4 = 20. (NOT 5 · 22 = 102 = 100.)
- (2 + 3)2 = 52 = 25. (Parentheses take precedence over powers.)
The Basic Rules of Exponents
There are three basic rules. Other rules will build upon these.
Now if powers stands for repeated multiplication, then what do negative powers mean? What about fractional powers? How do you multiply something by itself -3 times or 1/2 of a time?
The key is that we can interpret the expressions in ways that make the rules of algebra consistent.
For example, there is a rule that states “anything to the zero power is equal to 1.” Why should that be true? Well, it follows from the quotient rule for exponents:
1 = x1 / x1 = x1 – 1 = x0
In a similar way, you can interpret negative powers at fractions:
x – a = x0 – a = x0 / xa = 1 / xa
So, 5 – 3 = 1 / (53) = 1/125.
Note: A negative exponent does not mean that the result is a negative number!
Fractional powers are trickier. For example, we have a rule that states: . In other words, a fractional power is the same as a radical (or root)!
The reason for this strange rule is that it’s what we need to do to keep all of the fundamental rules consistent. If x1/2 equals something, then consider what happens when you square both sides:
So if something squared equals x, then that “something” must be the square root of x.
Similarly, if n is a whole number, then the fractional power x1/n is the same thing as the nth root of x.
All of the exponent rules for negative and fractional powers are a consequence of keeping things consistent algebraically. Here is the full list of the rules for zero, negative, and fractional powers.
The rules can be combined. For example, let’s work out 27-4/3.
Furthermore, there are algebraic rules that define how exponents and radicals interact with multiplication and division. These are very important when breaking down complicated expressions into simpler forms.
The following example will show how a few rules work together to simplify a very complex expression.
First combine the radicals using the product property.
Then combine factors. The coefficients should be multiplied together directly. But use the exponent rules to deal with the variables and powers.
Next, split the radical once again over the products. For the variable factors, it may help to rewrite the radicals as powers using the rules for fractional exponents.
The square root of 36 is 6. Multiply powers to simplify the variable factors. Finally, we have our simplified expression!
Now let’s see if we can apply what we’ve learned to the following GED Math practice problems! Answers and explanation will be given at the end. Good luck!
- Solve 64.
- A. 24
- B. 36
- C. 216
- D. 1,296
- A. -288
- B. -18
- C. 219
- D. 7,452
- A. 7x6
- B. 10x6
- C. 10x5
- D. 7x5
- A. -1
- B. -13
- C. 77
- D. 65
- A. 10
- B. 100
- C. 150
- D. 1000
- A. 6x2
- B. 3x
- C. 48x5
- D. 6x
Answers and Explanations
- Answer: D. A whole number exponent stands for repeated multiplication. 64 = 6 × 6 × 6 × 6 = 1,296.
- Answer: B. Follow the correct order of operations. Powers first. Then multiplication. Then addition and subtraction.
4 + 25 – 6 · 32
= 4 + 32 – 6 · 9
= 4 + 32 – 54
= 46 – 54
- Answer: C. Use the rules to rewrite the expression. Remember, when you multiply two factors that have the same base, you have to add the powers.
(2x3)(5x2) = (2 × 5) x3 + 2 = 10x5.
- Answer: A. First, plug in x = 2 and y = 1. Then work out the powers, following proper order of operations. Keep in mind that a negative exponent stands for a fractional value.
3(1)4 – 8(2) – 1 = 3 × 1 – 8 × (1/2) = 3 – 4 = -1.
- Answer: D. Here we have to interpret the fractional power as an appropriate radical.
- Answer: B. Pay special attention to the grouping. In the first group, the exponent applies to both the 2 and the x.