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# Exponent Properties with Variable Expressions

## Operations with exponents, including negative and fractional exponents

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Practice Exponent Properties with Variable Expressions
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Product and Quotient Properties of Exponents

Miguel says that the expression $\frac{2^5 \cdot 2^4}{2^2}$ equals $2^{10}$ .

Alise says that it is equal to $2^7$ .

Carlos says that because the exponents of the terms are different, the expression can't be simplified.

Which one of them is correct?

### Watch This

Watch the first part of this video, until about 3:30.

### Guidance

To review, the power (or exponent) of a number is the little number in the superscript. The number that is being raised to the power is called the base . The exponent indicates how many times the base is multiplied by itself.

There are several properties of exponents. We will investigate two in this concept.

#### Example A

Expand and solve $5^6$ .

Solution: $5^6$ means 5 times itself six times.

$5^6 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 15,625$

#### Investigation: Product Property

1. Expand $3^4 \cdot 3^5$ .

$\underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{3^4} \cdot \underbrace{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}_{3^5}$

2. Rewrite this expansion as one power of three.

$3^9$

3. What is the sum of the exponents?

$4 + 5 = 9$

4. Fill in the blank: $a^m \cdot a^n = a^{-^+-}$

$a^m \cdot a^n = a^{m+n}$

Rather than expand the exponents every time or find the powers separately, we can use this property to simplify the product of two exponents with the same base.

#### Example B

Simplify:

(a) $x^3 \cdot x^8$

(b) $xy^2 x^2 y^9$

Solution: Use the Product Property above.

(a) $x^3 \cdot x^8 = x^{3+8} = x^{11}$

(b) If a number does not have an exponent, you may assume the exponent is 1. Reorganize this expression so the $x$ ’s are together and $y$ ’s are together.

$xy^2 x^2 y^9 = x^1 \cdot x^2 \cdot y^2 \cdot y^9 = x^{1+2} \cdot y^{2+9} = x^3 y^{11}$

#### Investigation: Quotient Property

1. Expand $2^8 \div 2^3$ . Also, rewrite this as a fraction.

$\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2 \cdot 2}$

2. Cancel out the common factors and write the answer one power of 2.

$\frac{\cancel{2} \cdot \cancel{2} \cdot \cancel{2} \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{\cancel{2} \cdot \cancel{2} \cdot \cancel{2}} = 2^5$

3. What is the difference of the exponents?

$8 - 3 = 5$

4. Fill in the blank: $\frac{a^m}{a^n} = a^{-^--}$

$\frac{a^m}{a^n} = a^{m-n}$

#### Example C

Simplify:

(a) $\frac{5^9}{5^7}$

(b) $\frac{x^4}{x^2}$

(c) $\frac{xy^5}{x^6 y^2}$

Solution: Use the Quotient Property from above.

(a) $\frac{5^9}{5^7} = 5^{9-7} = 5^2 = 25$

(b) $\frac{x^4}{x^2} = x^{4-2} = x^2$

(c) $\frac{x^{10} y^5}{x^6 y^2} = x^{10-6} y^{5-2} = x^4 y^3$

Intro Problem Revisit Using the Product Property and then the Quotient Property, the expression can be simplified.

$\frac{2^5 \cdot 2^4}{2^2}\\= \frac{2^9}{2^2}\\= 2^7$

Therefore, Alise is correct.

### Guided Practice

Simplify the following expressions. Evaluate any numerical answers.

1. $7 \cdot 7^2$

2. $\frac{3^7}{3^3}$

3. $\frac{16x^4 y^5}{4x^2 y^2}$

1. $7 \cdot 7^2 = 7^{1+2} = 7^3 = 343$

2. $\frac{3^7}{3^3} = 3^{7-3} = 3^4 = 81$

3. $\frac{16x^4 y^5}{4x^2 y^2} = 4x^{4-2} y^{5-3} = 4x^2 y^2$

### Vocabulary

Product of Powers Property
$a^m \cdot a^n = a^{m+n}$
Quotients of Powers Property
$\frac{a^m}{a^n} = a^{m-n}; a \ne 0$

### Practice

Expand the following numbers and evaluate.

1. $2^6$
2. $10^3$
3. $(-3)^5$
4. $(0.25)^4$

Simplify the following expressions. Evaluate any numerical answers.

1. $4^2 \cdot 4^7$
2. $6 \cdot 6^3 \cdot 6^2$
3. $\frac{8^3}{8}$
4. $\frac{2^4 \cdot 3^5}{2 \cdot 3^2}$
5. $b^6 \cdot b^3$
6. $5^2 x^4 \cdot x^9$
7. $\frac{y^{12}}{y^5}$
8. $\frac{a^8 \cdot b^6}{b \cdot a^4}$
9. $\frac{3^7 x^6}{3^3 x^3}$
10. $d^5 f^3 d^9 f^7$
11. $\frac{2^8 m^{18} n^{14}}{2^5 m^{11} n^4}$
12. $\frac{9^4 p^5 q^8}{9^2 pq^2}$

Investigation Evaluate the powers of negative numbers.

1. Find:
1. $(-2)^1$
2. $(-2)^2$
3. $(-2)^3$
4. $(-2)^4$
5. $(-2)^5$
6. $(-2)^6$
2. Make a conjecture about even vs. odd powers with negative numbers.
3. Is $(-2)^4$ different from $-2^4$ ? Why or why not?

### Vocabulary Language: English

Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Power

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
power to a power

power to a power

Power to a power is a number raised to an exponent which in turn is raised to another exponent.
Product of Powers Property

Product of Powers Property

The product of powers property states that $a^m \cdot a^n = a^{m+n}$.
Quotients of Powers Property

Quotients of Powers Property

The quotient of powers property states that $\frac{a^m}{a^n} = a^{m-n}$ for $a \ne 0$.