We are given a table of vocabulary words.

Vocabulary Word
$1 .$ cube root
$2 .$ irrational number
$3 .$ Product of Powers Property
$4 .$ perfect cube
$5 .$ perfect square
$6 .$ Power of Powers Property
$7 .$ Powers of Products Property
$8 .$ scientific notation
$9 .$ square root

We are also given a table of definitions. Let's assign a letter to each definition.

Definition
A. a number that cannot be written in the form $\frac{a}{b},$ where $a$ and $b$ are integers and $b \neq = 0$
B. a way to express a number as the product of two factors, one greater than or equal to $1$ and less than $10,$ and the other a power of $10$
C. a number that when multiplied by itself equals the original number
D. the cube of an integer
E. a number whose cube equals the original number
F. To multiply two powers with the same base, keep the common base and add the exponents.
G. To multiply two powers with the same exponent and different bases, multiply the bases and keep the exponent.
H. a number that is the square of an integer
I. When you have an exponent raised to a power, keep the base and multiply the exponents.

Let's connect each vocabulary word with its definition. We will consider each word one at a time.

$1 .$ cube root

The cube root of a number is a number whose cube is equal to that number. For example, the cube root of

8

is equal to

2 .

38 = 3 2 \cdot 2 \cdot 2 = 2

This corresponds to definition E.

Vocabulary Word	Definition
$1 .$ cube root	E. a number whose cube equals the original number

$2 .$ irrational number

Numbers that are not rational are called irrational. An irrational number is a number that cannot be written in the form

\frac{a}{b},

where

a

and

b

are integers and

b \neq = 0 .

Let's take a look at a few examples of irrational numbers.

3, 5.39275832229 \dots, π

We can see that this corresponds to definition A.

Vocabulary Word	Definition
$2 .$ irrational number	A. a number that cannot be written in the form $\frac{a}{b},$ where $a$ and $b$ are integers and $b \neq = 0$

$3 .$ Product of Powers Property

Let's consider the product of two powers

3^{2}

and

3^{3} .

3^{2} \cdot 3^{3}

The number

3

is first raised to the second power and then to the third power. This means that

3

is multiplied two times and then three times.

3^{2} \cdot 3^{3} = (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3)

After expanding the powers, we can see that

3

is multiplied five times.

3^{2} \cdot 3^{3} = (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3) = 3^{5}

The Product of Powers Property states that when multiplying two powers with the same base, we add the exponents. This corresponds to definition F.

Vocabulary Word	Definition
$3 .$ Product of Powers Property	F. To multiply two powers with the same base, keep the common base and add the exponents.

$4 .$ perfect cube

Recall that the cube root of a number is a number whose cube is equal to that number. For example, the cube root of

27

is

3 .

327 = 3 3 \cdot 3 \cdot 3 = 3

A perfect cube is a number that is a cube of an integer. Therefore,

27

is a perfect cube. Other examples of perfect cubes are

8,

64,

and

125 .

2 \cdot 2 \cdot 2 = 8 4 \cdot 4 \cdot 4 = 64 5 \cdot 5 \cdot 5 = 125

We can see that this corresponds to definition D.

Vocabulary Word	Definition
$4 .$ perfect cube	D. the cube of an integer

$5 .$ perfect square

We already know that a perfect cube is a number that is a cube of an integer. The definition of a perfect square is similar. It is a number that is a square of an integer. A few examples of perfect squares are

4,

9,

16,

and

25 .

2 \cdot 2 = 4 3 \cdot 3 = 9 4 \cdot 4 = 16 5 \cdot 5 = 25

We can see that it corresponds to definition H.

Vocabulary Word	Definition
$5 .$ perfect square	H. a number that is the square of an integer

$6 .$ Power of Powers Property

Let's raise

2^{3}

to the second power.

(2^{3})^{2}

This means that we want to multiply

2^{3}

by

2^{3} .

(2^{3})^{2} = 2^{3} \cdot 2^{3}

Note that by the Product of Powers Property, we can keep the base and add the exponents.

(2^{3})^{2} = 2^{3} \cdot 2^{3} = 2^{6}

The Power of Powers Property states that to find the power of a power, we multiply the exponents. This corresponds to definition I.

Vocabulary Word	Definition
$6 .$ Power of Powers Property	I. When you have an exponent raised to a power, keep the base and multiply the exponents.

$7 .$ Powers of Products Property

Let's multiply two exponential expressions

4^{3}

and

2^{3} .

We will also expand the powers.

4^{3} \cdot 2^{3} = 4 \cdot 4 \cdot 4 \cdot 2 \cdot 2 \cdot 2

Now we can use the Commutative Property of Multiplication.

4^{3} \cdot 2^{3} = 4 \cdot 4 \cdot 4 \cdot 2 \cdot 2 \cdot 2 = (4 \cdot 2) \cdot (4 \cdot 2) \cdot (4 \cdot 2) = (4 \cdot 2)^{3}

The Powers of Products Property states that when multiplying two exponential expressions with the same exponent and different bases, we multiply the bases and keep the exponent the same. This corresponds to definition G.

Vocabulary Word	Definition
$7 .$ Powers of Products Property	G. To multiply two powers with the same exponent and different bases, multiply the bases and keep the exponent.

$8 .$ scientific notation

Scientific notation is a way to write very large or very small numbers. Scientists use scientific notation as a more efficient and convenient way of writing such numbers.

83000000 = 8.3 \times 1 0^{7} 0.0000456 = 4.56 \times 1 0^{- 5}

The definition of scientific notation is B.

Vocabulary Word	Definition
$8 .$ scientific notation	B. a way to express a number as the product of two factors, one greater than or equal to $1$ and less than $10,$ and the other a power of $10$

$9 .$ square root

The square root of a number is a number whose square is equal to that number. For example, the square root of

16

is equal to

4 .

16 = 4

This corresponds to definition C.

Vocabulary Word	Definition
$9 .$ square root	C. a number that when multiplied by itself equals the original number

1. cube root

2. irrational number

3. Product of Powers Property

4. perfect cube

5. perfect square

6. Power of Powers Property

7. Powers of Products Property

8. scientific notation

9. square root

Exercise

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