Intro to Exponents
1) Fundamentally, b^n means n factors of b multiplied together
We read 7^8 either as:
a) 7 to the power of eight, or
b) 7 to the eighth
Power of 2 is squared, power of 3 is cubed
2) one to any power is one
3) zero to any positive power is zero
4) a negative to an even power is positive; a negative to an odd power is negative
5) an equation with an expression to an even power equal to a negative has no solution, but an odd power can equal a negative
6) basic powers of the single digit numbers
The powers of 2 up to at least 2^9:
2,4,8,16,32,64,128,256,512
The power of 3 up to at least 3^4:
3,9,27,81
The power of 4 up to at least 4^4:
4,16,64,256
The power of 5 up to at least 5^4:
5,25,125,625
the cubes of 6-9:
6^3=216 7^3=343 8^3=512 9^3=729
Exponential Growth
How in easing the exponent changes the size of the powers for different kinds of bases
Case1: positive base greater than 1: the powers continually get larger, at a fast rate
Case2: positive base less than 1: the power continually to produce smaller powers
Case3: negative base less than -1: the absolute values are getting bigger each time, but the +/- signs are alternating
Case4: negative base btw -1 and 0: the absolute values are getting smaller, approaching zero, but the +/- signs are alternating
Law of Exponents - 1
1) (a^m)(a^n) = a^(m+n)
2) a^m/a^n = a^(m-n)
3) a^0 = 1 ( a=/= 0)
4) (a^m)^n = a^(m*n)
5) common exponents mistake
6) no exponent law for a sum or a difference of powers
Negative Exponents
1) b^(-n) = 1/b^n
2) a base to a negative exponent is the one over the base to the positive of that exponent
3) a fraction to the -n equals the reciprocal to the +n
4) exponents switch from negative to positive when we move powers in a fraction from the numerator to denominator or vice versa
Law of Exponents - 2
1) exponents distribute over multiplication and division
(ab)^n = (a^n)(b^n)
(a/b)^n = a^n/b^n
2) exponents do NOT distribute over addition and subtraction ((a+b)^n =/= a^n + b^n)
3) we can simplify the sum or difference of powers by factoring out the lower power
4) if a^m = a^n, then m=n
Units Digit Questions
Strategy for the units digit question:
1) focus on single-digit multiplication only
2) look for the repeating pattern, and determine the period of the pattern(period often is 4)
3) extend the pattern, using multiples of the period
Square Roots
1) the square sign always has a positive output
2) if we take the square root ourselves, we must consider both +/- roots
3) we covered some basic approximations for square roots
4) if b>1, then sqrt(b) < b.
5) if 0 < b <1, then sqrt(b) >b.
Other Roots
1) unlike with square root, we can take cube roots of both positives and negatives
2) in fact, we can take any even root of positives only, not negatives, but we can take any odd root of any number on the number line
3) any root of 1 equals 1, and any root of 0 equals 0
4) all roots preserve the order of inequalities (assuming all numbers are positive).
5) The higher the order of a root, the closer the result is to 1
Cubes up to 10 memorized
1,8,27,64,125,216,343,512,729,1000
If b>1, and if n>m, then 1<sqrt(b,n)<sqrt(b,m)<b
If 0<b<1, and if n>m, then 0<b<sqrt(b,m)<sqrt(b,n)<1
Properties of Roots
1) roots distribute over multiplication and division
sqrt(AB) = sqrt(A)sqrt(B)
sqrt(A/B) = sqrt(A)/sqrt(B)
2) roots do NOT distribute over addition and subtraction
sqrt(a) + sqrt(b) =/= sqrt(a+b)
sqrt(a) - sqrt(b) =/= sqrt(a-b)
Simplifying Roots
1) simplify square roots by factoring out the largest perfect-square factor
2) if we find the prime factorization, or are given it, then we can use that: any pairs of prime factors and any even powers of primes are perfect squares
Operations with Roots
1) when we add or subtract radical expressions, we simplify each term and combine terms with like radicals
2) when we multiply or divide radical expressions, we treat whole numbers and radicals separately, and we can multiply or divide two radicals right through the radical
3) when we raise a radical expression to a power, we distribute the exponent to each factor; any even power of a radical is a power of a whole number
Equations with Square Roots
1) to ‘undo’ a radical in an equation, we need to square both sides
2) we have to move something else to the other side to isolate the radical before squaring
3) the act of squaring produces extraneous roots. Therefore, we must check each answer the algebra gives us back in the original equation
Fractional Exponents
1) roots are represented by fractional exponents
2) a square root of a quantity = quantity to the power of 1/2
3) the power b^(1/n) means the nth root of b
4) the power b^(m/n) can be expressed either as the nth root of b^m(sqrt(b^m)) or as the nth root of b all to the power of m((sqrt(b,n))^m).
Exponential Equations
1) to solve exponential equations, we have to get equal bases on both sides. This may involve expressing the given bases as powers of smaller bases
2) once the bases on both sides are equal, we can equate the exponents and solve
Rationalizing
1) to eliminate roots from the denominator of a fraction is called rationalizing
2) if the fraction has a single root in the denominator, we rationalize by multiplying by that root over itself
3) if the denominator of the fraction contains Addison or subtraction involving a radical expression, to rationalize we need to multiply by the conjugate of the denominator over itself
Working with Formulas
1) the test will always give you the formula and explain what you need to know about it
2) the test may ask you to plug in numbers and solve, or it may ask you to solve for a variable in terms of the other variables
3) In a proportional reasoning question, it is usually easiest to make each number and the constant equal 1 in the starting case, and then change what needs to be changed
a) pick ridiculous easy numbers that satisfy the equation for a start. You also can change any constants to 1
b) change whatever values need to be changed, leave the quantity in the question as an unknown, and solve for it
