Law of Exponents

Law of Exponents

Objectives:

a. Apply the laws involving positive integral exponents to simplifying expressions involving zero and negative integral exponents; and

b. Solve problems involving positive and negative integral exponents and zero exponents.

Pre-Test!

Use the law of exponents to simplify each expression.

Multiple Choice

 $x^4\cdot x^7$  

•  $x^{11}$  

•  $x^{28}$  

•  $x^3$  

Multiple Choice

 $\left(p^4\right)^5$  

•  $p^9$  

•  $p^{21}$  

•  $p^{20}$  

Multiple Choice

 $\left(\frac{x}{y}\right)^8$  

•  $\frac{x^8}{y^8}$  

•  $\frac{x}{y^8}$  

•  $\frac{x^8}{y}$  

Multiple Choice

 $\frac{10m^8}{2m^3}$  

•  $5m^5$  

•  $\frac{5}{m^5}$  

•  $\frac{m^5}{2}$  

Multiple Choice

 $7x^0y^5z^0$  

•  $7x$  

•  $7y^5$  

•  $7xy^5z$  

Laws Involving Positive Integral Exponents to Zero

Lesson 3.1

Product of Powers

• To multiply powers with the same base, keep the base and add their exponents

• a^m * aⁿ = a^m+n

• Examples:

  
  

•  $2^2\cdot2^3\ =2^{2+3}\ =\ 2^5$  

•  $x^4\cdot x^5=x^{4+5}=x^9$  

•  $5y\cdot y^{10}=5y^{1+10}=5y^{11}$  

Multiple Choice

 $2x^2\cdot x^6$  

•  $2x^8$  

•  $x^8$  

•  $3x^8$  

Multiple Choice

 $2x^2\cdot x^6$  

•  $2x^8$  

•  $x^8$  

•  $3x^8$  

Power of a Power

• To find power of a power, keep the base and multiply the exponents

•  $\left(a^m\right)^n=a^{m\cdot n}=a^{mn}$  

• Examples:

•  $\left(3^2\right)^2=3^{2\cdot2}=3^4=81$  

•  $\left(a^2\right)^3=a^{2\cdot3}=a^6$  

Multiple Choice

 $\left(y^6\right)^2$  

•  $y^{16}$  

•  $y^{12}$  

•  $y^8$  

Multiple Choice

 $\left(x^6\right)^7$  

•  $x^{42}$  

•  $x^{13}$  

•  $x^4$  

Power of a Product

• A product raised to a power is equal to the product of its factors raised to the same power.

•  $\left(ab\right)^m=a^mb^m$  

• Examples:

•  $\left(mn\right)^2=m^2n^2$  

•  $\left(3a\right)^3=3^3a^3=27a^3$  

Multiple Choice

 $\left(4abc\right)^3$  

•  $64a^3b^3c^3$  

•  $12a^3b^3c^3$  

•  $16a^3b^3c^3$  

Multiple Choice

 $\left(2mnop\right)^4$  

•  $16m^4n^4o^4p^4$  

•  $8m^4n^4o^4p^4$  

•  $16m^4n^4p^4$  

Quotient of Power

• To divide powers with the same base, subtract their exponents.

•  $\frac{a^m}{a^n}=a^{m-n}\ ,\ a\ne0$  

• Examples:

•  $7^8\div7^5=7^{8-5}=7^3=343$  

•  $\frac{x^{12}}{x^8}=x^{12-8}=x^4$  

Multiple Choice

 $\frac{a^{45}}{a^{15}}$  

•  $a^{30}$  

•  $\frac{1}{a^{30}}$  

•  $a^{60}$  

Multiple Choice

 $\frac{15x^8y^3}{3x^2y^3}$  

•  $5x^6$  

•  $3x^6y^7$  

•  $5x^6y^7$  

Power of a Quotient

• When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power.

•  $\left[\frac{a}{b}\right]^m=\frac{a^m}{b^m}$  

• Examples:

•  $\left[\frac{x}{y}\right]^5=\frac{x^5}{y^5}$  

•  $\left[\frac{3}{n}\right]^2=\frac{3^2}{n^2}=\frac{9}{n^2}$  

Multiple Choice

 $\left[\frac{4}{x}\right]^2$  

•  $\frac{16}{x^2}$  

•  $\frac{8}{x^2}$  

•  $\frac{16}{x^1}$  

Multiple Choice

 $\left[\frac{3}{4y}\right]^2$  

•  $\frac{9}{16y^2}$  

•  $\frac{6}{8y^2}$  

•  $\frac{6}{8y}$  

Zero Exponent Rule

• For any nonzero number a, with an exponent of zero is equal to one.

•  $a^0=1,\ where\ a\ne0$  

• Examples:

•  $\left(5m\right)^0=5^0m^0=\left(1\right)\left(1\right)=1$  

•  $15x^0y^2=15\left(1\right)y^2=15y^2$  

Multiple Choice

 $\left[\frac{10000x^{5677}}{6675x^{2345}}\right]^0$  

• 1

•  $1234x^{3456}$  

• 0

Multiple Choice

 $\left(5x^0m^4\right)^{ }$  

•  $5m^4$  

• 1

• 0

Negative Exponent Rule

• For any nonzero number a and any integer n, a^-n is the reciprocal of aⁿ.

•  $a^{-n}=\frac{1}{a^n}\ and\ a^n=\frac{1}{a^{-n}}$  

• Examples 1:

•  $\frac{1}{5^{-2}}=\frac{1}{\frac{1}{5^2}}$  

•  $\frac{1}{1}\cdot\frac{5^2}{1}=\frac{5^2}{1}=5^2=25$  

Example 2:

•  $\frac{3^{-2}}{2^{-3}}=\frac{\frac{1}{3^2}}{\frac{1}{2^3}}$  

•  $\frac{1}{3^2}\cdot\frac{2^3}{1}=\frac{2^3}{3^2}=\frac{8}{9}$  

Multiple Choice

 $a^{-8}$  

•  $\frac{1}{a^8}$  

•  $a^8$  

•  $none\ of\ the\ choices$  

Multiple Choice

 $\frac{m^{-5}}{n^{-10}}$  

•  $\frac{n^{10}}{m^5}$  

•  $\frac{m^5}{n^{10}}$  

•  $\frac{m^{10}}{n^5}$  

Post Test

Read each statement carefully. Choose the correct answer.

Multiple Choice

Which of the following is equal to one?

• x^-1

• x⁰

• x¹

• $\frac{1}{x}$

Multiple Choice

What law of exponent is applied in (b²)(b⁵) = b⁷?

• Quotient of Powers

• Power of a Monomial

• Product of Powers

• Power of a Power

Multiple Choice

How do you you write the expression  $\frac{1}{m^{-8}}$  with positive exponents?

•  $m^8$  

•  $m^0$  

•  $m$  

•  $\frac{1}{m^8}$  

Multiple Choice

What is the simplest form of the expression  $2^{-5}$  using positive exponents?

• 32

• 10

•  $\frac{1}{32}$  

• -32

Multiple Choice

Which of the following is equivalent to  $\frac{a^{-6}}{a^{12}}$  ?

•  $a^6$  

•  $a^{18}$  

•  $\frac{1}{a^6}$  

•  $\frac{1}{a^{18}}$  

"Life is a polynomial equation with a leading exponent of positive infinity, so you will never run out of solutions."

- Norain