Unlock the Secrets of Perfect Squares, Cubes, and Powers: Base and Exponent

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

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
$1$	$2$	$2$
$2$	$2 \cdot 2$	$4$
$3$	$2 \cdot 2 \cdot 2$	$8$

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 Square?

Explanation

25

5 \times 5 = 5^{2}

Yes

✓

5

is an integer.

30.25

5.5 \times 5.5 = 5 . 5^{2}

\times

5.5

is not an integer.

32

5.656 \dots \times 5.656 \dots = (5.656 \dots)^{2}

\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.

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}

\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}

\times

6.463 \dots

is not an integer.

a A perfect square is a number that can be expressed as the square of an integers. For example, the number

64

can be written as the square of

8 .

8^{2} = 64

Since

8

is an integer,

64

is a perfect square. To identify the numbers that are not perfect squares, consider the squares of

5

and

6 .

5^{2} = 25 6^{2} = 36

Since

5

and

6

are integers,

25

and

36

are perfect squares. There are no integers between

5

and

6 .

This means that there are no perfect squares between

25

and

36 .

It can be noted that $27$ is not a perfect square. Now, to determine if $136$ is a perfect square, remember that the square of $10$ is $100 .$ The consecutive perfect squares are displayed in the table below.

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

As shown in the table, $136$ is greater than $121$ but less than $144 .$ Since $136$ lies between two consecutive perfect squares, $136$ cannot be a perfect square. More perfect squares will be identified to determine if $225$ is a perfect square.

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

The number $225$ is the square of $15 .$ Since $15$ is an integer, $225$ is a perfect square. The remaining number is $729 .$ The square of $20$ is $400 .$ The square of $30$ is $900 .$ Since $729$ is closer to $900$ than to $400,$ it might be convenient to start with $30$ and explore the next perfect squares in decreasing order.

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

Notice that $729$ is a perfect square. Now that all numbers have been examined, the results can be summarized in another table.

Number	Perfect Square?
$64$	Yes
$27$	No
$136$	No
$225$	Yes
$729$	Yes

b A perfect cube is a number that can be expressed as the cube of an integer. For example, the cube of

10

1000 .

Therefore,

1000

is a perfect cube.

10^{3} = 1000

Since all the given numbers are less than

1000,

any perfect cubes among them can be expressed as the cube of an integer from

2

9 .

Make a table to write these cubes and look for the given numbers, starting with

27 .

Number	Cube	Less Than, Greater Than, or Equal to $27 ?$
$2$	$2^{3} = 8$	$8 < 27$
$3$	$3^{3} = 27$	$27 = 27$

Because $27$ can be expressed as the cube of the number $3,$ it is a perfect cube. Using a similar reasoning, the number $64$ will be examined.

Number	Cube	Less Than, Greater Than, or Equal to $64 ?$
$4$	$4^{3} = 64$	$64 = 64$

As shown, $64$ is a perfect cube. Now continue the table to see if $136$ is a perfect cube.

Number	Cube	Less Than, Greater Than, or Equal to $136 ?$
$5$	$5^{3} = 125$	$125 < 136$
$6$	$6^{3} = 216$	$216 > 136$

The number $136$ lies between two consecutive perfect cubes, so $136$ is not a perfect cube. Now the number $225$ will be tested. It is a good thing that $216$ is really close to $225!$

Number	Cube	Less Than, Greater Than, or Equal to $225 ?$
$6$	$6^{3} = 216$	$136 < 216$
$7$	$7^{3} = 343$	$343 > 216$

The number $225$ lies between the perfect cubes $216$ and $343,$ meaning that $225$ is not a perfect cube. Finally, $729$ will be tested.

Number	Cube	Less Than, Greater Than, or Equal to $729 ?$
$8$	$8^{3} = 512$	$512 < 729$
$9$	$9^{3} = 729$	$729 = 729$

Every number was examined. The results are summarized in the table below.

Number	Perfect Cube?
$64$	Yes
$27$	Yes
$136$	No
$225$	No
$729$	Yes

Notice that a number can be both a perfect cube and a perfect square!

Exponent	Result
Even	Always positive
Odd	Sign depends on the base

Exponent

Result

Even

Always positive

Odd

Sign depends on the base

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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

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