The following is everything I have learned about exponents in the past few weeks.

 

1. Repeated Multiplication With Exponents

All exponents can be written 4 ways: In word from, repeated multiplication, exponential form, and evaluated.

2. How Powers Represent Repeated Multiplication

The two basic parts of an exponent are the base and the exponent (power). The base is the number being multiplied by itself, and the exponent or power is the amount of times it is multiplied.

So, if we are writing this out as repeated multiplication, the number 3 multiplied by itself 4 times would look like this:

3) Demonstrate The Difference Between The Exponent And Base

The exponent and base are different and cannot be interchanged. This can be proven by making a model of and comparing it to a model of . This will give us shapes of completely different sizes.

Notice how in one model represented volume, while the other was area.

As we can see, the two exponents gave us two completely different models. It is important not to mix up the base and exponent. As well, it is important to note that (2x2x2) does not equal 2×3.

 

4) Interchanging The Exponent And Base

For this example, we’ll use the exponents and . We will provide more proof that these do not evaluate to the same number by using repeated multiplication.

5) Evaluating Powers With Integral Bases

We will go through a few examples of solving (evaluating) exponents via repeated multiplication with bases that are positive and negative.

As shown, negative bases behave in the same way as positive ones. The only difference is the product will be negative if the power is an odd number.

Another thing important to mention is that if there is a number being multiplied by the base, it is called the coefficient.

6) The Role Of Parentheses

Brackets can play a major role in finding the answer for an exponent. Their placement can greatly affect the outcome:

*Note: The last two examples are identical, aside from the second one having (orange) brackets that allow it to be further used in another expression.

7) Product And Quotient Laws

To make our lives easier, when we are multiplying or dividing powers of the same base, there are some shortcuts we can take. Keep in mind, this only applies if the bases are the same number. 

In this example, it is shown that when multiplying exponents with the same base, you can simply add the powers and have one single exponent that you can then easily evaluate. This is the product law.

Here, we can see that to make a division expression simpler, we can subtract the exponents (not to be confused with adding them in a multiplication question). This is the quotient law.

These two shortcuts will make simplifying expressions with powers much quicker and simpler. However, there are no rules such as these for addition and subtraction. For those operations, you need to evaluate the exponents first and then add/subtract them.

We can also look at an expression in which there are coefficients.

The same rules apply, but all we add is simplifying the coefficients.

8) Raising A Product Or Quotient To An Exponent

To simplify more complex expressions, in which there are exponents inside and outside brackets, we can apply the power law. Simply put, this law tells us to multiply the inside exponent by the outside one.

Notice how we applied the power before the product or quotient laws. As a rule of thumb for complex equations, first use BEDMAS, then the power law, and finally the product and quotient laws in the order they appear (left to right).

9) 0 As An Exponent

The rule for 0 as an exponent is very simple. Anything to the power of 0 will always equal one. Even if the base is a negative number. The reason (-2)º isn’t (-1) is because we know that an even amount of negatives cancels out and makes a positive. 0 is an even number, so the product/quotient must be positive.

However, problems occur when you bring 0 to the power of 0. There’s a theory that shows when finding , we see that the smaller x gets, the nearer the answer gets to 1. Perhaps, 0º is infinitely close to 1 but not quite there, but it is safe to call it “undefined”.

10) Using Patterns To Prove xº = 1

We can use a simple pattern of bringing 2 to different powers to prove that anything to the power of 0 equals 1.

As we can see, every time the power decreases by 1, the evaluated number gets divided by the base.

11) Negative Powers

There are a few things to keep in mind when evaluating numbers with negative powers. To remove the negative power, you can reciprocal the base and the exponent can become positive.

When evaluating with fractions, the same rule applies, although there is a less confusing way to write it:

The first strategy went through the same process as with whole numbers, but as you can see, we ended up with a fraction in a fraction. The second (and preferred) strategy reciprocals the fraction and changes the power to a positive. This still gives the same answer, but is a much more simple and correct strategy.

12) Patterns Explaining Negative Power Law

The concept above may have been confusing, but it is easily explained by continuing a pattern we started when evaluating powers of 0.

We simply continued dividing by the base to get smaller and smaller fractions, proving that the above strategy works.

13) Applying Exponent Laws

In this section we will apply the laws we have learned to exponents with integral and variable bases. We’ll start with a simple expression with both positive and negative powers. It is important to remember that in order to change a negative power to a positive one, we must move the exponent to the other side of the fraction.

 

In this example we first applied the power law to the top of the fraction, then the quotient law to the top and product law to the bottom, and finally we sorted out the negative power. Make sure to always deal with negatives at the very end.

Let’s look at an expression with some variables:

Here, we didn’t have to worry about the power law. However, notice how we still left negative exponents until the very end, where we turned them positive by placing them on the opposite side of the fraction.

14) Finding Errors In The Simplification Of An Expression

Not following the order of operations and laws can give incorrect solutions.

In this example, the first equation was solved incorrectly by assuming the two negative powers cancelled out. Then, using the quotient law, the answer was achieved.

But that was incorrect! 2 negative powers do not cancel out. As well, we apply all rules for negative powers last. So instead, we should first use the quotient rule to get a single negative power, then place it as a denominator in a fraction and turn it positive. It’s important to do these steps in the correct order to get the right solution.

15) Order Of Operations With Powers

In the simple example provided, we show that it is always necessary to follow the rules of BEDMAS. Had we not, the answer would have been 36, which is definitely not equal to 12.

16) Sums And Differences

We know that there are shortcuts for multiplying and dividing exponents, but for addition and subtraction, we have to go the long way

.

In this example, we also included a coefficient. According to the order of operations, we know that we have to do what is in the brackets first. So we figure out the exponents, then add them, then multiply the coefficient.

 

In this expression, we showed that it is important to first figure out the exponents, then use the rest of BEDMAS to simplify the rest of the expression.

17) Errors In Applying Order Of Operations

Clearly, there have been many mistakes made in the first solution of this equation. Firstly, the brackets were not completed first, but the power was applied to the numbers.Remember, in BEDMAS, brackets always go first. Secondly, the base was multiplied by the coefficient before being evaluated to the power, which again breaks the order of operations laws.

18) Use Powers To Solve Problems (Measurement)

Exponents can be used to solve many types of problems, such as measuring area or volume.

While we may have been able to do these questions previously in our heads, writing them out as exponents can really help when getting into more complex problems. Problems including area and volume only differ by whether you make your exponent 2 or 3, representing the number of dimensions you are working with.

19) Solving Growth Problems

Here, we have a word problem involving growth or doubling over a certain amount of time. In this case, Larry’s bitcoin doubles in cost every month. In one month, that’s 1¢ x 2. The following month, we have to take that initial equation and multiply it by 2 again. We could write it as 1 x 2 x 2, or use exponents as a shortcut and write 1 x 2². The 1 doesn’t do anything in our expression, but we should keep it there because if it was any other number, it would change our product. I chose a 1 just for simplicity when calculating. So, to continue our pattern, after the third month, we would have 1 x 2³. Notice the correlation between the exponent and the duration of time. This means we can take any number of months and use it as the exponent in our expression. This is shown with the last example, where our variable, x, can be any number. All we need to do is use it as our exponent and we have the answer!

20) Negative Exponents And Variable Bases

Let’s take a look at one final expression to summarize what we’ve learned.

To keep it short, I did multiple steps at once in the places that are highlighted. Note that the cº became 1, which was added to 2 to make a sum of 3.

As well, when the whole expression is to be raised to a negative exponent, we can flip it to make the exponent positive. If the expression was not a fraction but instead a simple multiplication question, we would place it as the denominator in a fraction where the numerator is 1.

Finally, after applying the power, product, and quotient laws, we were able to get rid of all negative powers by moving the exponents to the other side of the fraction. Keep in mind, the final answer can be negative, as long as none of the powers are.

20.5) Additional Information

We can also go through how to solve simple equations where the power is a variable.

To start off, we should re-write the equation so the bases match.

Next, we can get rid of the bases, because if they are equal, the exponents must be equal.

Now, we can easily solve this equation with some basic math.

This is the basis for solving more complex equations with variable powers, and as well, it concludes this post of everything I know about exponents.