Exponents

What are exponents?

• Exponents are the number of times the base is being multiplied by itself, in the image above the power shown is 3² which can be written as 3×3. For example, if you look at the power of 4⁵ it can be written as 4x4x4x4x4 and 5 is the exponent because that’s how many times 4 is being multiplied by itself. The base is the number being multiplied, in the power 6³ the base is 6. The power is the whole thing including the base and the exponent.

2. How do powers represent repeated multiplication?

– Exponents are repeated multiplication because an exponent is how many times the base is multiplied by itself, so when writing powers using repeated multiplication it is just expanding the expression. This is a more visual way to understand how exponents work and can be helpful because it will help ensure you don’t make the mistake of multiplying the base by the exponent or adding/forgetting to place a negative sign. It is also more simplified and shorter to write version, the exponent and the repeated multiplication still mean the same thing.

4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2³ and 3².

– The reason these two numbers could be confused is that when doing the expression you may multiply the base and the exponent, however when simplifying an expression like that you need to multiply the base by the base x amount of times. So when these two expressions are simplified 2³ = 8 and 3² = 9. 2³ and 3² ≠ 6. Another mistake that you might make is assuming that 2³ and 3² are equal but the base and the exponent are not interchangeable.

– Another way to identify the difference between the two powers is to use a model like squares and cubes, this way you can clearly see that 3² ≠ 2³

– This example is not the only one that can be complicated, for example sometimes when simplifying an expression like 3³ can be confused as 3×3 which equals 9 however 3³ = 27 because 3x3x3=27. One way to correct this is to keep in mind that the exponent is not being multiplied but shows how many times the base will be multiplied by itself.

6. Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)⁴, (-2⁴) and -2⁴.

– If the exponent is on the outside of the brackets and the base has a negative attached (such as (-2)⁴) then if the exponent is odd it will be negative, if the exponent is even it will be positive. This is because you are multiplying the base WITH the negative because it’s in brackets. Therefore (-2)⁴= (+16) but (-2)⁵= (–32).

– If the exponent is in the inside of the brackets and the base has a negative next to it (such as (-2⁴)) then it will always be a negative answer. This is because the negative is not connected to the base, in the expression the negative acts as a coefficient. It may be easier to imagine the single negative as -1 because they both equal the same thing. Therefore (-2⁴) = (–16) and (-2⁵) = (–32), any number you input in either of those spots that is of larger value then 0 will equal a negative.

– If the exponent has no brackets and the base has a negative next to it (such as -2⁴) it will always be a negative answer for the same reasons as the explanation above. This is because -2⁴ and (-2⁴) are the same thing and will equal the same thing with the exception that -2⁴ when simplified will not have the answer in brackets.

Also, you can see if the simplified version of power will be positive or negative based on the number of negative symbols. If there are an odd amount the answer will be negative, but if there are an even amount the answer will be negative. For example (-3)² has an even amount of negatives because (-3)² = (-3)(-3) and so you can see that this would be positive so (-3)²=9.

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

– In an expression like 2⁵x2² where both bases are the same you add the exponents so 2⁵x2²=2⁷. One way to prove this is to write it out in total 2⁵x2² would equal (2)(2)(2)(2)(2)x(2)(2) = 2⁷

– If the bases are the same then you subtract the exponents, similar to how you add them for products, in the expression 2⁵÷2² you would keep the base and subtract the exponents meaning it would equal 2³. If it’s in a fraction you use the same rules above to apply exponents, and, if the bases are the same you subtract the exponents. One way to prove this rule is to solve it first, for example, 2⁵=32 and 2²=4 so 32÷4=8 and 8 = 2³.

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

– The pattern displayed in the photo can be applied to any base, further proving that anything to the exponent of 0 equals 1 as long as the base ≠ 0. This pattern helps to identify not only exponent of 0 but it also helps to more deeply understand how exponents function. Therefore even if you were to use an algebraic expression such as x, and have it written to the exponent of 0 you would know that it would equal 1.

12. Use patterns to explain the negative exponent law.

– To understand negative exponents using patterns you can use the same method for finding anything to the exponent of zero. When following the pattern of exponents into the negatives you can clearly see that the rule still applies and the amount the number is being divided by never changes. This visual tool helps to understand why negative exponents are written as fractions.

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

• Common errors that can occur during the simplification of an expression with powers are multiplying the exponent by the base, using incorrect exponent laws, not properly applying BEDMAS and incorrectly interpreting brackets. Most of these problems have been addressed above however it is important to go over them quickly. Multiplying the exponent by the base is incorrect because the exponent represents how many times the base is multiplied by itself. When applying exponent laws to ensure that you are using the proper one, product law applies to the multiplication of powers when the base is the same, quotient law applies to the division of powers if the base is the same and power law applies when there are multiple powers on a single base. BEDMAS is used whenever powers are being added or subtracted because you cannot apply exponent laws to sums or differences, BEDMAS is also used when the bases are not the same and you cannot use product and quotient laws. Brackets are a very important part of an expression because if there is a negative it can mean the difference between the answer being a positive or a negative, they also indicate which part of the problem to address first and can separate multiple exponents that are influencing the same base.

16. Determine the sum and difference of two powers.

• When adding or subtracting powers you cannot use product, quotient or power laws. Instead, you apply BEDMAS (brackets, exponents, division, multiplication, addition, and subtraction) to solve. So a^x+bⁿ = axa x amount of times + bxb n amount of times. The same applies to subtraction.

18. Use powers to solve problems (measurement problems)

• Squares: when adding or subtracting anything to the exponent of two (squared) you can use squares to represent them in a more visual way. To solve it if you are given the length of one side and need to find the area you just square the side length to get the area if you are subtracting or adding one square with another you have to calculate each area and then add or subtract those areas.
• Cubes: when adding or subtracting anything to the exponent of three (cubed) you can use cubes to represent them in a more visual way. This also explains (for both squares and cubes) how exponents apply to measurements. To solve it if you are given the length, width or height and need to find the volume you just cube the side length to get the volume if you are adding or subtracting one cube from another you have to calculate each respective volume and add or subtract those volumes.
• Pythagorean Theorem: Pythagorean theorem is a²⁺b²=c² and is used to find the hypotenuse. This formula contains exponents and can be drawn out as squares and a triangle. If you are given two variables (a, b or c) you are able to calculate the remaining variable.
• Inscribed Circles: it is a circle that is inscribed inside of a square. To find the area of the circle you have to take the side length of the square and divide it by two to get the radius from there you use the formula πr² to get the area of the circle. From here you calculate the total area of the square by squaring the side length. Next, subtract the area of the circle from the total area of the square to find the area of the square that is not being occupied by the circle.

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

• Order of operations or BEDMAS is the order in which a mathematical problem has to be completed. This is important to remember when working with powers because the solution is dependent upon BEDMAS.
• Variables are usually letters (such as x, y, n, a, b, etc.) that are used to represent an unknown value.

What else do I know about exponents?

What is the power law?

• The power law is used when there are multiple exponents being applied to a base, such as (3²)³. When you have an expression like this you multiply the two exponents, therefore (3²)³=3^2×3=3⁶
• Exponent laws for raising a product to an exponent is simple, if for example, you have an expression like (2×3)³ then you apply the exponent to both numbers inside of the brackets so (2×3)³ will look like 2³x3³ because we are applying the exponent to each number.
• Exponent laws for raising a quotient to an exponent is also quite easy, but in this case, there are two ways that this can be observed. It can be in a basic written form or a fraction if for example, you have an expression like (2÷3)³ than just like raising the product to an exponent you apply the exponent to both numbers inside of the brackets to make the problem look like 2³÷3³.

3 exponent laws:

1. Product law states that if the base is the same you keep the base and ADD the exponents
2. Quotient law states that if the base is the same you keep the base and SUBTRACT the exponents
3. Power law states that you keep the base and MULTIPLY the exponents

Patterns in Exponents:

• When solving exponents with integral bases you can identify patterns in the last digit. These patterns can be visually represented through a listing of the exponents and their simplified versions like the examples above. This is consistent if the base and exponent ≠ 0.

Link to Regan’s Edublog Post(odds): https://myriverside.sd43.bc.ca/reganb2019/2019/11/08/everything-i-know-about-exponents/

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