Powers

Looking at the product, we can see that 7 is being multiplied five times. 7* 7* 7* 7* 7_5 We need to determine which number is the base and which is the exponent to write the power expression. The base is the number being multiplied and the exponent is the number of times that the base is multiplied. Let's identify which is which! Base: & 7 Exponent: & 5 Now that we know which number is the base and which is the exponent, we can write the power that corresponds to this product of repeated factors. 7* 7* 7* 7* 7_5 ⇕ 7^5

Looking at the product, we can see that this time 11 is being multiplied seven times. 11* 11* 11 * 11 * 11 * 11 * 11_7 Similar to what we did before, we will identify the base and the exponent. Now that we know how to do it, let's write them! Base: & 11 Exponent: & 7 Now that we know which is the base and which is the exponent, we can write the power for this product. 11* 11* 11 * 11 * 11 * 11 * 11_7 ⇕ 11^7

Expression	Example $1$	Example $2$
$2^{2}$	$2$ to the second power	$2$ squared
$7^{3}$	$7$ to the third power	$7$ cubed
$5^{4}$	$5$ raised to the power of $4$	$5$ raised to the fourth power

Questions Answered Correctly	Points	Simplify
$1$	$2$	$2$
$2$	$2 \cdot 2$	$4$
$3$	$2 \cdot 2 \cdot 2$	$8$

Example	Rewrite as a Product	Perfect Square?	Explanation
$25$	$5 \times 5 = 5^{2}$	Yes $✓$	$5$ is an integer.
$30.25$	$5.5 \times 5.5 = 5 . 5^{2}$	No $\times$	$5.5$ is not an integer.
$32$	$5.656 \dots \times 5.656 \dots = (5.656 \dots)^{2}$	No $\times$	$5.656 \dots$ is not an integer.
$64$	$8 \times 8 = 8^{2}$	Yes $✓$	$8$ is an integer.

Example	Rewrite as a Product	Perfect Cube?	Explanation
$125$	$5 \times 5 \times 5 = 5^{3}$	Yes $✓$	$5$ is an integer.
$166.375$	$5.5 \times 5.5 \times 5.5 = 5 . 5^{3}$	No $\times$	$5.5$ is not an integer.
$64$	$4 \times 4 \times 4 = 4^{3}$	Yes $✓$	$4$ is an integer.
$270$	$6.463 \dots \times 6.463 \dots \times 6.463 \dots = (6.463 \dots)^{3}$	No $\times$	$6.463 \dots$ is not an integer.

Number	Square	Less Than, Greater Than, or Equal to $136 ?$
$10$	$1 0^{2} = 100$	$100 < 136$
$11$	$1 1^{2} = 121$	$121 < 136$
$12$	$1 2^{2} = 144$	$144 > 136$

Powers

Catch-Up and Review

How to Write Many Multiplications

Powers, Exponents, and Bases

Rewriting Products

Bonus Points in a Trivia Contest

Hint

Solution

Evaluating Powers

Raising a Number to the Power of $2$

Perfect Square

Raising to the Power of $3$

Perfect Cube

Testing for Perfect Squares and Perfect Cubes

Hint

Solution

Identifying Perfect Squares and Perfect Cubes

Even and Odd Powers

Powers

Recommended exercises

	11 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Number	Square	Less Than, Greater Than, or Equal to $225 ?$
$13$	$1 3^{2} = 169$	$169 < 225$
$14$	$1 4^{2} = 196$	$196 < 225$
$15$	$1 5^{2} = 225$	$225 = 225$

Number	Square	Less Than, Greater Than, or Equal to $729 ?$
$30$	$3 0^{2} = 900$	$900 > 729$
$29$	$2 9^{2} = 841$	$841 > 729$
$28$	$2 8^{2} = 784$	$784 > 729$
$27$	$2 7^{2} = 729$	$729 = 729$

Number	Square	Less Than, Greater Than, or Equal to 325?
20	20^2 = 400	400 > 325
19	19^2 = 361	361 > 325
18	18^2 = 324	324 < 325

Number	Square	Less Than, Greater Than, or Equal to 1158?
30	30^2 = 900	900 < 1158
31	31^2 = 961	961 < 1158
32	32^2 = 1024	1024 < 1158
33	33^2 = 1089	1089 < 1158
34	34^2 = 1156	1156 < 1158
35	35^2 = 1225	1225 > 1158

Number	Square	Less Than, Greater Than, or Equal to 2809?
50	50^2 = 2500	2500 < 2809
51	51^2 = 2601	2601 < 2809
52	52^2 = 2704	2704 < 2809
53	53^2 = 2809	2809 = 2809

Number	Square	Less Than, Greater Than, or Equal to 3249?
60	60^2 = 3600	3600 > 3249
59	59^2 = 3481	3481 > 3249
58	58^2 = 3364	3364 > 3249
57	57^2 = 3249	3249 = 3249

Number	Cube	Less Than, Greater Than, or Equal to 2197?
10	10^3 = 1000	1000 < 2197
11	11^3 = 1331	1331 < 2197
12	12^3 = 1728	1728 < 2197
13	13^3 = 2197	2197 = 2197

Number	Cube	Less Than, Greater Than, or Equal to 2864?
13	13^3 = 2197	2197 < 2864
14	14^3 = 2744	2744 < 2864
15	15^3 = 3375	3375 > 2864

Exponent	Result
Even	Always positive
Odd	Sign depends on the base

Number	Cube	Less Than, Greater Than, or Equal to 526?
9	9^3 = 729	729 > 526
8	8^3 = 512	512 < 526

Number	Cube	Less Than, Greater Than, or Equal to 343?
8	8^3 = 512	512 > 343
7	7^3 = 343	343 = 343