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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.

James Sousa: Properties of Exponents

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}

Answers

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?

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