Everything I know about exponents

1. Represent repeated multiplication with exponents

Repeated multiplication is the base multiplied by the base, however many times the exponent dictates.

Ex: is the same as 5x5x5x5x5x5.

 

2. Describe how powers represent repeated multiplication

A power is however many times the base is multiplied by the base

Ex: means the four is being multiplied by four, six times.

 

3. Demonstrate the difference between the exponent and the base by building models of a given power, such as and .

The difference between these two powers is significant. The exponents are different which means that the base is multiplied a different amount of times.

Ex: is 3×3 and is 2x2x2.

 

4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as and

The difference is that ‘s base is being multiplied twice whereas  ‘s base is being multiplied three times.

Ex:  = 3×3=9 = 2x2x2=8

5. Evaluate powers with integral bases (excluding base 0) and whole number exponents.

All bases(excluding zero) can be put to a power. For instance, you could do .

Ex: is okay but doesn’t. Any exponent to a base of zero is undefined.

6.  Explain the role of parentheses in powers by evaluating a given set of powers such as , , .

Depending on where you have the parentheses, it will change your answer completely. As you can see below, those are all very different questions.

Ex: = (-2)(-2)(-2)(-2)

=(-1x2x2x2x2)

= -1x2x2x2x2

7. Explain the exponent laws for multiplying and dividing powers with the same base.

When multiplying powers with the same base, all you have to do is keep the base and add the exponents together. When dividing, you have to keep the base and subtract the exponents. These are called the product law and the quotient law.

Ex: x =

Ex: =

8. Explain the exponent laws for raising a product and quotient to an exponent.

When raising a product or quotient to an exponent you put however many times the base is multiplied in the place of the exponent.

Ex: 5x5x5x5x5 =

9.  Explain the law for powers with an exponent of zero.

When you have the exponent of zero, it will equal one, regardless of the base. It could be one or one million.

Ex: = 1 (this will work for all bases with the power of zero except zero)

10. Use patterns to show that a power with an exponent of zero is equal to one.

To show that when there is a power of zero, you can look at the quotient rule and division. We know that 7 7 =1. Quotient rule says that you subtract the exponents.   is . Since those are the same, we can conclude that the power of zero is 1.

Ex: =1 = =1

11. Explain the law for powers with negative exponents.

When you have a negative exponent such as what you have to do is turn it into its reciprocal. That way you can turn it into a positive exponent.

Ex: =

12. Use patterns to explain the negative exponent law.

Like I said before, you need to turn the expression into it’s reciprocal just like turning 4 into . The only difference is that the denominator will have an exponent.

Ex:   will turn into =

13.  I can apply the exponent laws to powers with both integral and variable bases.

There really isn’t much of a difference between the two types of bases. When you have an integral base, you multiply the base the number of times the exponent dictates and that’s it. When you have a variable base (such as x), you can’t calculate it since it’s a variable so it stays the same.

Ex: =64     =

14. I can identify the error in a simplification of an expression involving powers.

Yes, I can and it’s not that hard. I know the rules I need to follow what some common mistakes to look for are.

Ex: someone may think it’s but it’s actually five times the product of .

15. Use the order of operations on expressions with powers.

It is not difficult, you just have to pay attention and follow BEDMAS.

Ex: x = 64×81= 5,184 .

No Brackets, Exponents first, no Division, Multiplication second, no addition and no subtraction.

16. Determine the sum and difference of two powers.

Unlike when multiplying and dividing numbers with powers, when adding and subtracting, you solve for the power and then add or subtract those two sums.

Ex: + = 64+25= 89 . The same rule applies with subtraction.

17. Identify the error in applying the order of operations in an incorrect solution.

Making an error in the order of operations can make a big difference. For example…

Ex: =20

They did 4×5 instead of 4x4x4x4x4 which is 1,024 which is a big difference.

18. Use powers to solve problems (measurement problems)

Using powers to solve these problems is actually quite easy and effective especially when dealing with squares and cubes. Here are some good examples,

Ex: You have a square with the side length of 5m. What is the area? We know that a square has four equal sides so taking that information, we can deduce that it would be 5×5 or . From there it’s easy and we can figure out that the area is 25.

19.  Use powers to solve problems (growth problems)

Using powers to solve these types of problems is also very effective. A great example is bacteria.

Ex: If you have 2 bacteria and they multiply by two once every 30mins, how many bacteria would there be at… 1 hour, 6 hours, 12 hours…

First of all, you can figure out that it multiplies twice in one hour. From there you can multiply that (2) by the number of hours you have. So if we apply it to six, 2×6 is 12 so that would be . 12 hours would be 2×12 which is 24 or double that of six hours.

 

1 hour =

6 hours =

12 hours =

20. Applying the order of operations on expressions with powers involving negative exponents and variable bases.

With this, it’s actually quite simple. you just have to follow one rule which is flip everything upside-down! Just kidding but that’s part of it. Think of it as a giant division and the numbers get divided but because the variables can’t, the variables get put on the opposite side from where they are (refer to picture example1and 2). If they aren’t they just stay where they are. If there are any numbers with negative exponents, they get turned into their reciprocal and then dealt with.

Ex:          (pic ex1)            (pic ex2)                 This image gives a very good example of what one of these questions might look like and how they might be solved. It also shows the swapping of sides I talked about. As you can see, everything gets turned into a positive exponent. You then apply the quotient law and voila! you are done.