In algebraic expressions, we will use a combination of letters and numbers. Letters are often referred to as variables, since the value can change. We may also call these unknowns, if it is something we are asked to solve for. Numbers are constants because their value will never change.
When multiplying an unknown with a number, they can be written next to each other. For example: 4 * x = 4x. So, when you see a number with a letter attached to it, you know the number will be multiplied by the value of the unknown. If we know x = 5, then 4x = 4(5) = 20. When we have an equation with a letter attached to the number, that number is called a coefficient.
Let's consider the following expression:
If we know that x = -2 and m = -3, we can substitute those values for the respective letters in the expression. To help ensure proper signs are retained and the order of operations is not affected, we can use parentheses around the values as we plug them in. For this example, we'll do this one step at a time. If we replace x with -2, we get:
Notice, the original expression said x times m. With the value of x plugged into the expression, we are still multiplying. It now reads -2 times m. Now we can replace the m unknowns with -3:
Now we can use the order of operations to simplify the expression into a single value:
When we have a coefficient outside of a set of parentheses with multiple terms inside the parentheses, we have to use the distributive property to simplify the expression. This means that the number in front of the parentheses is multiplied by each term inside the parentheses. Consider the following example:
2 is the coefficient. x and -1 are the terms inside the parentheses. To distribute the 2, we multiply it by each term in the expression. When we do this, we can also remove the parentheses.
We now have two terms: 2x and -2. We cannot simplify this expression any further because these are not like terms. The x makes the term 2x not compatible with a term that does not also contain x. Let's consider another example:
We will distribute the -3 across both terms inside the parentheses:
Notice that the x term becomes negative since the coefficient was negative and the integer was positive. The constant becomes positive because both the coefficient and the constant were negative. (negative times negative equals positive)
What if we have two expressions being multiplied together? Applying the distributive property, each term in each expression must be multiplied by every term in the other expression. That's a lot to conceptualize, so let's consider a visual:
Following the red arrows, we are distributing the x term by multiplying it by the 2x and 3 terms in the other parentheses. Doing just that half, we get:
We will combine that with the second half in just a moment, so hang on to it. Let's follow the blue arrows to distribute the -1:
Now we can put those together before we simplify further. Be sure to keep the same signs:
When writing algebraic expressions that include unknowns, we want to order them by the value of their exponent. Thus, x-squared comes before x. Now we can combine like terms. Note: x-squared is not a like term with x, so we cannot combine them. We can combine the two x terms though.
This would be the final simplification. Let's say we know that x = 2. Let's plug that in and solve:
When we do repeated multiplication, or take some value times itself, we can rewrite that using exponents. The base is the value being multiplied, and the exponent is the number of times it is being multiplied by itself. When the base is a letter, it is called an unknown.
In this instance, 4 is multiplied by itself 2 times. That is 4 x 4. Likewise, if we take x times x, we get x-squared:
Some terms you might see associated with exponents:
It is important to note the value of the base when evaluating exponents. Consider the following:
At first glance, these may appear to be saying the same thing. However, the first expression is actually saying -1 times 3 squared. The second expression is -3 squared. When worked out, these would appear as follows:
What does it look like when we add, subtract, multiply, or divide terms with exponents?
When adding terms with exponents, we add the coefficients. Terms without a number in front of the unknown base are assumed to have a coefficient of 1 (because multiplying anything by 1 does not change its value):
Note: We must have common unknowns and exponents in order to combine terms through addition. These examples cannot be combined because they are not like terms:
or
The first expression contains two different unknowns, so we cannot combine those terms. The second expression contains the same unknown raised to different powers or exponents. Therefore, we cannot combine these terms either.
Subtracting terms with exponents follows the same rules as addition: 1) we will subtract the coefficients of the terms, 2) the unknowns must be the same letter, and 3) the exponents of the unknowns must be the same.
When multiplying terms with exponents, we add the exponents:
If you break apart each term, we can see why adding the exponents is appropriate. For instance:
and
therefore
When dividing terms with exponents, we will subtract the exponents. The placement of the remaining term will depend on the location of the larger exponent. Consider the following examples:
=
=
= 4f
and
Notice when the larger exponent was in the numerator, the simplified unknown remained in the numerator. However, when the larger exponent was in the denominator, the simplified unknown was also in the denominator.
Negative Exponents
Exponents can also take on negative values. You may be wondering, "How do I multiply something times itself a negative number of times?" That's a great question! The answer is a bit simpler. That is, a negative exponent indicates that the term is the reciprocal of its positive term. As always, let's consider some examples:
and
Mathematical operations with negative exponents follow the same rules as those stated above. Let's look at an example of each using a negative exponent:
Addition:
Subtraction:
Multiplication:
Division:
Notice in the last example how we could rewrite the final answer using the inverse of the unknown. You can always move the unknown to make the work simpler if that makes more sense to you. As long as it's done correctly, it will not affect the final outcome.
Now that you've learned about exponents, let's learn about the opposite of an exponent. You might be familiar with these as "square roots," but a more general term is radical. A square root is a specific type of radical that applies to squares (a value raised to the power of 2). Taking the square root of a number will tell you what number multiplied by itself gives you the number under the radical. For example:
We can also have roots with different powers, like a cube root. A cube root tells you what number times itself three times gives you the number under the cube root. For instance:
We can do this with any root value, but you typically won't see anything past a cube root in an Algebra course.
Simplifying Radicals
When we are simplifying radicals, we are considering the factors of the term under the radical and seeing if any can be "pulled out" perfectly. For square roots, we can pull out perfect squares. For a cube root, we can pull out perfect cubes. It is beneficial to memorize perfect squares and perfect cubes. It will make it easier to recognize them and think in terms of them when considering factors. Let's look at an example:
We could start by listing all of the factors of 135, that is, the numbers that could be multiplied to make 135, but that won't help us much here. What we really need to consider is: Does 135 have a factor that is a perfect square? Memorizing the perfect squares will help with this process! Until then, you can work through them in order using a calculator until you get to one that works. In this case, you'll find that 9 goes into 135 evenly:
So, we can rewrite the expression under the radical as
Since 9 is a perfect square, we can pull it out from under the radical. When we do this, we are taking the square root of that number. The square root of 9 is 3, because 3 times itself equals 9. We can then rewrite the expression as
This means we would be taking 3 times whatever the square root of 15 is. If you use a calculator, you can get a precise estimate (depending on how many digits your calculator can show). But, in Algebra, we often only reduce to its simplest form, keeping the radical in the expression.
