Everything I Know About Exponents

1) Represent repeated multiplication with exponents:

4 x 4 x 4 x 4 = $4^4$

The base of the power is what we multiply with, and the exponent is how many times we multiply the base by itself.

2) Describe how powers represent repeated multiplication:

$6^3$ = 6 x 6 x 6

The base would tell us what we multiply with and the exponent shows us how many times we multiply it by itself.

3) Demonstrate the difference between the exponent and the base by building models of a given power, such as $2^3$ and $3^2$ :

$2^3$ = 1st Image

This image shows: Length x Width x Height (Since it’s cubed)

$3^2$ = 2nd Image

This image shows: Length x Height (Since it’s squared)

The difference between these two models is, the the first model is 2 cubed (volume) = $2^3$ and the second model is showing a model for area which is $3^2$ and because the base and exponents look similar but the are quite different because of how the base and exponents relate to each other.

4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as $2^3$ and $3^2$ :

$2^3$ = 2 x 2 x 2

$3^2$ = 3 x 3

Well, the difference between these two powers are that the first one has a base of 2 and an exponent of 3 which means, the base multiplies by itself how many times the exponent tells it to; and the second power has a base of 3 and an exponent of 2 which means, the base is 3 and it has to multiply by itself 2 times because the exponent is 2.

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

Well some examples for evaluating powers would be when I choose a base and then an exponent so that I know how many times I multiply the base by itself:

$3^3$ = 27

$5^2$ = 25

$8^9$ = 134,217,728

$1^{10}$ = 1

$(-4)^4$ = 256

$(-7)^3$ = -343

$-1^{15}$ = -1

6) Explain the role of parentheses in powers by evaluating a given set of powers such as $({-2})^4$ , $({-2}^4)$ and  ${-2}^4$ :

Well in these powers there are “invisible” coefficients, and with these you have to pay attention to where and how the parentheses are placed. With these three powers, two of them are the same and the third would be different, for example:

$({-2})^4$ = (-2) (-2) (-2) (-2) = 16

$({-2}^4)$ = (-1)(2)(2)(2)(2) = (-16)

 ${-2}^4$ = (-1) x 2 x 2 x 2 x 2 = (-16)

7) Explain the exponent laws for multiplying and dividing powers with the same base.
The exponent law for multiplying powers with the same base is:

Keep the base
Add the exponents
Multiply coefficient

Two examples:

$3^2 \cdot 3^3$ = $3^{2+3}$ = $3^5$ = 243

$(2)3^2 \cdot (6)3^3$ = $(2 \cdot 6)$ $(3^{2+3})$ = $12 \cdot 3^5$ = $12\cdot 243$ = 2,916

The exponent law for dividing powers with the same base is:

Keep the base
Subtract exponents
Divide coefficients

Two examples:

$5^5 \div 5^3$ = $5^{5-3}$ = $5^2$ = 25

$(12)2^9 \div (6)2^7$ = $(12\div6)$ $2^{9-7}$ = $2\cdot2^2$ = $2\cdot4$ = 8

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

The exponent power law is:

Keep the base
Multiply the exponents
Raise coefficient to the power

Two examples:

$(3^5)^2$ = $(3^{5\cdot2})$ = $3^{10}$ = 59,049

$[(7)(6^2)]^2$ = $[(7^2)(6^{2\cdot2})]$ = $[(7^2)$ $(6^4)]$ = [(49)(1,296)] = 63,504

9) Explain the law for powers with an exponent of zero:

The zero law is:

Any power with an exponent of zero equals to ONE

Examples:

$3^0$ = 1

$7^0$ = 1

$453,298^0$ = 1

$(-2)^0$ = 1

$latex 0^0 ≠ 1

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

A pattern using a base of 2:

$2^5$ = 32

$2^4$ = 16

$2^3$ = 8

$2^2$ = 4

$2^1$ = 2

$2^0$ = 1

This pattern shows that $2^0$ = 1 and each power with an exponent of 0 = 1 because each time the exponents decreases by 1 you would divide by the base to get the answer, so following this pattern that is how we can show that any power with 0 as an exponent = to 1. BUT don’t forget this pattern can be shown with different bases and answers, but the pattern would still show the exact same thing, that each time you decrease (by 1) the exponent with the same base, you would divide by the base which would eventually show that any base raised to the power of 0 = 1 EXCEPT $0^0$ ≠ 1.

11) Explain the law for powers with negative exponents:

Any base, except 0 raised to a negative exponent equals to:

Reciprocal the base
Make exponent positive.

For example:

$4^{-3}$ = $\frac {1}{4^3}$ = $\frac {1}{64}$

12) Use patterns to explain the negative exponent law:

An example as a pattern:

$4^5$ = 1,204

$4^4$ = 256

$4^3$ = 64

$4^2$ = 16

$4^1$ = 4

$4^0$ = 1

$4^{-1}$ = $\frac {1}{4^1}$ = $\frac{1}{4}$

$4^{-2}$ = $\frac {1}{4^2}$ = $\frac{1}{16}$

$4^{-3}$ = $\frac {1}{4^3}$ = $\frac{1}{64}$

$4^{-4}$ = $\frac {1}{4^4}$ = $\frac{1}{256}$

$4^{-5}$ = $\frac {1}{4^5}$ = $\frac{1}{1,204}$

This pattern shows a base with a negative exponent and how it progresses from a positive exponent to negative and how you have to make sure the exponent is negative or positive to know if you should reciprocal it or not. BUT remember you ALWAYS have to reciprocal the base if the exponent is negative, and that’s how you get these answers with a one as the numerator in the fraction, and a positive exponent as the denominator. And you can observe the pattern of how the the answer is divided by the base to get the next answer if the exponent decreases by one, so this pattern goes on even if the exponents are negative.

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

Of course, this is possible, here are some examples:

$2^{5} \cdot 2^1$ = $2^{5+1}$ = $2^6$ = 64

$3^{11}\div3^7$ = $3^{11-7}$ = $3^4$ = 81

$5^{-2} \cdot 5^{-3}$ = $5^{-2+-3}$ = $5^{-5}$ = $\frac{1}{5^5}$ = $\frac{1}{3,125}$

$4^{-7}\div4^{-8}$ = $4^{-7+8}$ = $4^1$ = 4

$(x^2)^3 \cdot (x^7)^3$ = $(x^{2\cdot3})\cdot (x^{7\cdot3})$ = $x^6 \cdot x^{21}$ = $x^{6+21}$ = $x^{27}$

$(z^2)^{-6} \div (z^4)^{-1}$ = $(z^{2\cdot-6}) \div (z^{4\cdot-1})$ = $z^{-12} \div z^{-4}$ = $z^{-12+4}$ = $z^{-8}$ = $\frac{1}{z^8}$

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

One example:

$[7 (y^6)]^3 \cdot [5(y^3)]^3$ = $[7\cdot3(y^{6\cdot3})]$ $[5\cdot3(y^{3\cdot3})]$ = $[21(y^{18})]\cdot[15(y^9)]$ = $(21\cdot15)(y^{18+9})$ = $315y^{27}$

This is all incorrect because one simple mistake was made, the coefficients weren’t cubed they were just multiplied but the raised power, so this is one mistake that is quite easy to make while simplifying to get the answer. The correct answer would have been: $latex 42875y^{27}$

15) Use the order of operations on expressions with powers:

Using BEDMAS:

$4\cdot6^2\div 12$

= $4\cdot 36\div 12$

= $144\div12$

=12

16) Determine the sum and difference of two powers.

Using BEDMAS for addition:

$5^3 + 3^2$

= 125 + 9

=134

Using BEDMAS for subtraction:

$6^2 - 3^4$

= 36 – 81

= -45

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

Using BEDMAS incorrectly:

$[(198 + 2)\div 5]\cdot3^2 - 6$

= $[(198 + 2)\div 5]\cdot9 - 6$

= $[(198 + 2)\div 5]\cdot3$

= $(200\div5)\cdot3$

= $40\cdot3$

= 120

The error was not following BEDMAS, the correct way would be to first use the brackets, instead I chose to do the exponent and the subtraction part first which was the problem in this expression, if I were to do this correctly using BEDMAS I would have received the answer 354.

18) Use powers to solve problems (measurement problems):

One example:

Doms’ square locker has a square binder in it. The square binder has an area of $25cm^2$ his square locker has an area 10 times the area of the binder. What is the side length of the locker?

$\sqrt{25cm^2\cdot10}$

= $\sqrt{250cm^2}$

≈15.81cm

Doms’ locker has the approximate side length of 15.81cm in total.

19) Use powers to solve problems (growth problems):

An example:

Jim loves pets, he was able to buy an animal shelter, when he bought his first pet. Every hour the amount of pets double. How many are there after 1 hour? 3 hours? 8 hours?

1 hour: $2^1\cdot 1$ = 2 Pets.

3 hours: $2^3\cdot 1$ = 8 Pets.

8 hours: $2^8\cdot 1$ = 256 Pets.

Jims’ animal shelter doubles the amount of pets each hour so in 1 hour there would be two, in three hours there would be 3, and in 8 hours there would be 256, and so and so on.

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

An example:

$[3^{-3}]\cdot [5(x^{-2})]$

= $[\frac{1}{3^3}]\cdot[5\cdot \frac{1}{x^2}]$

= $\frac{1}{27}\cdot \frac{5}{x^2}$

= $\frac{5}{27x^2}$

2 comments

Maria says:

November 8, 2016 at 8:52 PM

This post really gave a better understanding of exponents and each example gives me a clear showing for what each question is asking for and how to act in each problem and how to receive the correct answer. Overall, this has really gave me better and deeper understanding of this unit and I believe you have done a great job!

alhanz2016 says:

November 11, 2016 at 4:34 PM

Overall you’ve addressed everything well and your examples are easy to understand. I don’t think I spotted any mathematical errors and you’ve done well in explaining everything. I can’t really pinpoint anything that could be improved, so good job.

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Everything I Know About Exponents

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