Algebra Ch 7: Exponents and Exponential Functions

Power of a Product Example

(-2xy³)⁵ = (-2)⁵x⁵y¹⁵ or -32x⁵y¹⁵ (ab) to the m = a to the m times b to the m

Power of a Quotient Example

(3/5)⁴ = 3⁴/5⁴ (r/t)⁵ = r⁵/t⁵ (a/b) to the m = a to the m over b to the m

Power of a Power Example

(b³)⁵ = b³ x ⁵ or b¹⁵ (g⁶)⁷ = g⁶ x ⁷ or g⁴² (a to the m) to the p = a to the m x p

What are the 3 different ways of solving the problem 8 to the 2/3

(³√8)² = 2² = 4 ³√8² = ³√64 = 64 ¹/³ = 4 2³ x ²/³ = 2² = 4

Word: Simplify Expressions Question: To simplify a monomial expression, write an equivalent expression in which does what?

1. each variable base appears exactly once, 2. there are no powers of powers, and 3. all fractions are in simplest form

Give Examples of Monomials

1.2, -420, 5/3, m, x, 5x, 5ab, 2x²

Zero Exponent Property Example

15⁰ = 1 (b/c)⁰ = 1 (2/7)⁰ = 1

geometric sequence

1st term is non-zero and each term after is found by multiplying the previous term by a non-zero constant r.

what understood number is typically on a square root sign

2

Negative Exponent Property Example

2 to the -4 = 1/2⁴ = 1/16 1/j to the -4 = j⁴

negative exponent example:

2/x to the -3 = 2x³

cube root example

2³=8 2= ³√8 8 to the 1/3 Symbol: a = ³√b

nth root example

2⁴ = 16, 2 is a fourth root of 16; ⁴√16 = 2 a to the n = b, then n√b = a

b to the 1/n example

8 to the 1/3 = ³√8 = 3√2x2x2 or 2

What is the equation for Compound Interest?

A = P(1 + r/n ) to the 'nt'

exponential function

A function that can be described by an equation of the form y = ab to the x, where a≠0, b > 0, and b≠1

What does each variable mean in the equation for Compound Interest?

A is the current amount P is the principal or initial amount n is the number of times the interest is compounded each year, and t is time in years. r is the annual interest rate expressed as a decimal, r > 0.

Zero Exponent Property

Any nonzero number raised to the zero power is equal to 1.

Power of a Product

Find the power of each factor and multiply.

Power of a Quotient

Find the power of the numerator and the power of the denominator.

b to the 1/2

For any nonnegative real number b, b to the 1/2 = √b EX: 16 to the 1/2 = √16 or 4 EX: 38 to the 1/2 = √38

Negative Exponent Property

For any nonzero number 'a' and any integer 'n', 'a' to the '-n' is the reciprocal of 'a' to the 'n'.

b to the 1/n

For any positive real number b and any integer n > 1, b to the 1/n = n√b

Power Property of Equality

For any real number b > 0 and b ≠ 1, b to the x = b to the y if and only if x=y. EX: If 5 to the x = 5³, then x = 3. If n = 1/2, then 4 to the n = 4 to the 1/2

nth root

For any real numbers a and b and any positive integer n, if a to the n = b, then a is an _______ of b

What is the simple interest formula?

I=prt

Power of a Power

Multiply the exponents.

Quotient of Powers

To divide two powers with the same base, subtract the exponents.

Product of Powers

To multiply two powers that have the same base, add their exponents

Exponential Decay Functions

When an initial amount decreases by the same percent over a given period of time.

Exponential Growth Functions

When an initial amount increases by the same percent over a given period of time.

order of magnitude example

Your Calculation =75 Your Final Answer= 10²; 100

What does each variable mean in the equation for exponential decay?

a is the initial amount y is the final amount t is time r is the rate of decay expressed as a decimal, 0 < r < 1.

constant

a monomial that is a real number

monomial

a number, a variable, or the product of a number and one or more variables with non-negative exponents - it has only one term + and - signs separate these - division is multiplication of a fraction

scientific notation formula

a x 10 to the n 1 ≤ a < 10

exponential equation

an equation with variable exponents - b to the x

rational exponent

an exponent that is a rational number

zero exponent

anything to the zero power is 1

What is the recursive arithmetic formula?

a₁ = 1, aₙ = aₙ-₁ + d, n ≥ 2

What is the recursive geometric formula?

a₁ = 1, aₙ = r x aₙ-₁, n ≥ 2

What is the explicit arithmetic formula?

aₙ = a₁+ (n-1) d

What is the equation for a geometric sequence?

aₙ = a₁rⁿ-¹

What is the explicit geometric formula?

aₙ = a₁rⁿ-¹

in the problem √b what is b read as.

b is read as the radicand

How do you write out √b?

b to the 1/2 the numerator is the power and the denominator is the root.

What is an example of a rational exponent?

b to the 1/2 = √b

Product of Powers Example

b³ x b⁵ = b³+ ⁵ or b⁸ g⁴ x g⁶ = g⁴+ ⁶ or g¹⁰ a to the m x a to the p = a to the m+p

Quotient of Powers Example:

c⁵/c² = c⁵-² or c³ a to the m over a to the p = a to the m-p

what variable does common difference use and what type of sequence is it?

d; arithmetic

How do you figure out if a problem is geometric?

division of the 2nd term by the 1st

geometric is what type of function?

exponential

what is another name for a ratio?

fraction

compound interest

interest earned or paid on both the initial investment and previously earned interest.

What does an explicit problem mean?

it can solve for any number

arithmetic is what type of function?

linear

negative exponent

move the exponent to the denominator and make it positive. If the exponent is negative in the denominator, you move it to the numerator and make it positive.

Non-negative exponent

no variable in denominator

common ratio

r from the geometric sequence - can be found by dividing any term by its previous term.

what variable does common ratio use and what type of sequence is it?

r; geometric

a square root sign (√) has another name that is called what?

radical sign

How do you figure out if a problem is arithmetic?

subtraction

When an exponent crosses the division bar, what happens?

the sign switches.

order of magnitude

used to compare measures and to estimate and perform rough calculations - round the number to the nearest power of 10

scientific notation

when a number is to large or small to write out so you use this.

cube root

when a³ = b, a is the _________ of b.

Give Examples that are not Monomials

x to the -3 power, 1/x³

What is the power, base and exponent of x⁵

x⁵ - power x - base 5 - exponent (power)

exponential function examples

y = 2(3) to the x y = 4 to the x y = (1/2) to the x

What is the equation for exponential growth

y = a(1+r) to the t

What is the equation for exponential decay?

y = a(1-r) to the t

What does each variable mean in the equation for exponential growth?

y is the final amount a is the initial amount t is time r is the rate of change expressed as a decimal, r > 0.

What does an geometric problem mean?

you need to know the number before the last number to get the final answer