24.1 Properties of Exponents Know the meaning of exponent, zero exponent and negative exponent.Know the properties of exponents.Simplify expressions involving exponentsKnow the meaning of exponential function.Use scientific notation.
3Exponent For any counting number n, We refer to as the power, the nth power of b, or b raised to the nth power.We call b the base and n the exponent.
4Examples When taking a power of a negative number, if the exponent is even the answer will be positiveif the exponent is odd the answer will be negative
5Properties of Exponents Product property of exponentsQuotient property of exponentsRaising a product to a powerRaising a quotient to a powerRaising a power to a power
6Meaning of the Properties Product property of exponentsRaising a quotient to a power
7Simplifying Expressions with Exponents An expression is simplified if:It included no parenthesisAll similar bases are combinedAll numerical expressions are calculatedAll numerical fractions are simplifiedAll exponents are positive
8Order of Operations Parenthesis Exponents Multiplication Division AdditionSubtraction
9WarningNote: When using a calculator to equate powers of negative numbers always put the negative number in parenthesis.Note: Always be careful with parenthesis
16Exponential Functions An exponential function is a function whose equation can be put into the form:Where a ≠ 0, b > 0, and b ≠ 1.The constant b is called the base.
17Exponential vs Linear Functions x is a exponentx is a base
18Scientific Notation A number written in the form: where k is an integer and-10 < N ≤ -1 or1 ≤ N < 10Examples
19Scientific to Standard Notation When k is positive move the decimal to the rightWhen k is negative move the decimal to the leftmove the decimal 3 places to the rightmove the decimal 5 places to the left
20Standard to Scientific Notation if you move the decimal to the right, then k is positiveif you move the decimal to the left, then k is negativemove the decimal 4 places to the leftmove the decimal 9 places to the right
23Rational Exponents ( ) For the counting number n, where n ≠ 1, If n is odd, then is the number whose nth power is b, and we call the nth root of bIf n is even and b ≥ 0, then is the nonnegative number whose nth power is b, and we call the principal nth root of b.If n is even and b < 0, then is not a real number.may be represented as
24Examples ½ power = square root ⅓ power = cube root not a real number since the 4th power of any real number is non-negative
25Rational ExponentsFor the counting numbers m and n, where n ≠ 1 and b is any real number for which is a real number,A power of the form or is said to have a rational exponent.
34Base Multiplier Property For an exponential function of the formIf the value of the independent variable increases by 1, then the value of the dependent variable is multiplied by b.
35x increases by 1, y increases by b -31/8-21/4-11/212438xy-1841231/2
36Increasing or Decreasing Property Let , where a > 0.If b > 1, then the function is increasinggrows exponentiallyIf 0 < b < 1, then the function is decreasingdecays exponentially
37Interceptsy-intercept for the form:is (0,a)is (0,1)
38Intercepts Find the x and y intercepts: y-intercept x-intercept as x increases by 1, y is multiplied by 1/3.infinitely multiplying by 1/3 will never equal 0as x increases, y approaches but never equals 0no x-intercept exists, instead the x-axis is called the horizontal asymptote
39Reflection Property The graphs are reflections of each other across the x-axisa > 0a > 0a < 0a < 0