Exponents and Exponential Functions

Exponents and Exponential Functions

A monomial is one term with a number, a variable or numbers and variables multiplied together, with nonnegative integer exponents.

​Monomials

​Coefficient = The number in front of a variable.

Base = The variable, number or expression that is being multiplied according to the variable

Exponent = The power or the number of times the base is being multiplied be itself

​

Parts of a Monomial

space

MULTIPLY the coefficients

and

ADD the exponents

when the bases are the same!

​

​Multiplying Monomials

Multiple Choice

$$b^3\cdot b^5=$$

• $$b^8$$

• $$b^{15}$$

• $$b^{-2}$$

• $$8b$$

When a monomial is raised to a power...

Apply the exponent to the coefficient

and

MULTIPLY the exponents !

​

Power of a Power

Multiple Choice

$$\left(x^5y^7\right)^4$$

• $$x^9y^{11}$$

• $$x^{20}y^{28}$$

• $$x^1y^3$$

• $$4x^54y^7$$

Multiple Choice

$$\left[\left(-2xy^2\right)^3\right]^2$$

• $$64x^6y^{12}$$

• $$-2x^6y^{12}$$

• $$64x^5y^7$$

• $$-2x^6y^{10}$$

​Divide the coefficients

and

subtract the exponents!

Dividing Monomials

​​Divide the coefficients

and

subtract the exponents!

Dividing Monomials

Multiple Choice

$$\frac{9a^5b^2c^8}{3ab^2c^3}$$

• $$\frac{9}{3}a^5b^4c^{11}$$

• $$3a^6b^4c^{11}$$

• $$\frac{3ab^2c^3}{9a^5b^2c^8}$$

• $$3a^4c^5$$

​A zero exponent on any base equals one.

​​Zero Exponents

A negative exponent on any base is the reciprocal of that monomial.

​​Negative Exponents

Multiple Choice

$$\frac{\left(3x\right)^0}{2a}$$

• $$\frac{3}{2a}$$

• $$1$$

• $$\frac{1}{2a}$$

• $$\frac{x}{2a}$$

Multiple Choice

$$\frac{12y^{-4}}{3x^{-5}}$$

• $$\frac{4x^5}{y^4}$$

• $$\frac{3x^5}{12y^4}$$

• $$4x^5y^4$$

• $$\frac{12x^5}{3y^4}$$

When an exponent is a fraction, the expression can be written as a radical expression.

​

​Rational Exponents

Multiple Choice

Write as a radical expression and then simplify:

$$625^{\frac{1}{4}}$$

• $$\sqrt[4]{625}=156.25$$

• $$\sqrt[4]{625}=5$$

• $$\frac{625}{4}=156.25$$

• $$4\left(\frac{1}{625}\right)=\frac{4}{625}$$

Multiple Choice

Write as a radical expression and simplify:

$$32^{\frac{2}{5}}$$

• $$2\left(\sqrt[5]{32}\right)=4$$

• $$\left(\sqrt[2]{32}\right)^5=4$$

• $$\left(\sqrt[5]{32}\right)^2=4$$

• $$5\left(\sqrt[2]{32}\right)=4$$

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Rewrite each side of the equation so they have the same base and then set the exponents equal to each other. Solve for x if necessary.

​

Solving exponential equations

Multiple Choice

$$5^x=125$$

• $$x=3$$

• $$x=25$$

• $$x=\frac{1}{3}$$

• $$x=5$$

Multiple Choice

$$2^{x-1}=32$$

• $$x=-4$$

• $$x=4$$

• $$x=3$$

• $$x=5$$

Write a very large number or very small number as a coefficient times 10 to an exponent. The coefficient should have one nonzero digit to the left of the decimal.

​

​Scientific & Standard Notation

Multiple Choice

Write in scientific notation:

$$2,300,000$$

• $$2.3^6$$

• $$2.3\times10^5$$

• $$2.3$$

• $$2.3\times10^6$$

Multiple Choice

Write in standard form:

$$5.43\times10^{-5}$$

• $$543,000$$

• $$0.0000543$$

• $$0.00543$$

• $$54300$$

​Functions with a variable in the exponent. Make a table to graph them. Then name the y-intercept, the domain (x values) and the range (y values) of the graph and identify if it is growth or decay.

​

Exponential Functions

Multiple Choice

Name the y-intercept, the domain and range, and tell whether it is growth or decay.

• y-intercept: (0,1)

  Domain: ℝ

  Range: y>0

  Decay

• y-intercept: (0,3)

  Domain: y>0

  Range: ℝ

  Decay

• y-intercept: (0,1)

  Domain: ℝ

  Range: y<0

  Growth

• y-intercept: (0,3)

  Domain: ℝ

  Range: y>0

  Growth

​​Growth Formula

​​Decay Formula

• a = initial value

• r = growth rate

• x = time

• n = number of times it is compounded

​​Compound Interest

Formula

​Exponential Growth & Decay

• a = initial value

• r = growth rate

• x = time

• a = initial value

• r = growth rate

• x = time

Multiple Choice

A new car that sells for $18,000 depreciates 25% each year. Write a function that models the value of the car. Find the value of the car after 4 yr.

• $$f\left(x\right)=18000\left(1-.25\right)^x$$

  $$f\left(4\right)=5695.31$$

• $$f\left(x\right)=18000\left(1+.25\right)^x$$

  $$f\left(4\right)=43945.31$$

• $$f\left(x\right)=18000\left(1-25\right)^x$$ $$f\left(4\right)=5695.31$$

• $$f\left(x\right)=25000\left(1-.18\right)^x$$

  $$f\left(4\right)=11303.04$$

In a geometric sequence, each term is multiplied by a common ratio r to get the next term in the sequence.

​

Geometric Sequences

Multiple Choice

Find the common ratio and the next three terms in this sequence:

1, 3, 9, . . .

• r = 1/3

  18, 27, 36

• r = 1/3

  27, 81, 243

• r = 3

  27, 81, 243

• r = 3

  12, 15, 18

Multiple Choice

Find the common ratio and the next three terms in this sequence:

64, 16, 4, . . .

• r = 1/2

  2, 1, 1/2

• r = 1/4

  1, 1/4, 1/16

• r = 1/4

  1, 1/4, 1/8

• r = 4

  1, 1/4, 1/8

​Formula to find any term in a sequence

• a_n = The term you are finding

• a₁ = The 1^st term in the sequence

• r = The common ratio

• n = The number of the term you are finding

​

Multiple Choice

Use the formula $$a_n=a_1r^{n-1}$$ to find the 10th term in the following sequence:

1, 3, 9, . . .

• 19683

• 90

• 729

• 6561

Multiple Choice

Use the formula $$a_n=a_1r^{n-1}$$ to find the 20th term in the following sequence:

64, 32, 16, . . .

• 2/25

• 33,554,432

• 0.000122

• 1/20