Chapter 5: Roots and Exponents

how do you multiply or divide numbers in scientific notation?

multiply or divide the coefficients separately from the powers of 10 (multiply the coefficients separately and then multiply the powers of 10 separately)

radical’s index

the number found on the top left of the radial

for even indexed roots n√x^n =

|x|

perfect square

when a whole number is multiplied by itself to produce another whole number

every perfect square is _______ except 0

positive

if the cube root of N is y, then N=

y^3

true or false: the cube root can be positive or negative

true

when simplifying radicals, apply the rule: √a x b =

√a + √b

when multiplying radicals that share the same index, combine them under

the same radical

when dividing radicals that share the same index, combine them under

the same radical

how would you solve a problem when dividing two perfect roots?

first take the perfect square root of each number separately and then divide the results

how do you solve problems when multiplying or dividing expressions that contain both radicals and non-radicals

multiply and divide radicals by radicals (if they have the same index) and non-radicals by non-radicals

when adding or subtracting terms under a square root, all operations inside the radical must be completed…

before taking the root

when adding or subtracting radicals with the same root index and the same radicand (value under the radical symbol), you can treat the radicand like a

variable and combine the two terms

how do you solve a fraction when the denominator has a binomial?

multiple the numerator and denominator by the conjugate

conjugate

it is formed by changing the sign of the second term

when taking the nth root of a number, if n is even, then n√x^n =

|x|

when taking the nth root of a number, if n is odd then n√x^n =

x

when taking an even root of a binomial raised to an even power

√(x+y)² =

|x+y|

how do you solve an equation with a square root and a variable under the radical?

isolate the radical, then square both sides of the equation to remove the radical, then solve

if a^x = a^y then

x=y

when multiplying two exponent expressions with the same base, keep the base and _____ the exponents

add

(x^a)(x^b)=

x^a+b

when dividing two exponential expressions with the same base, keep the base and ________ the exponents

subtract

x^a / x^b =

x^a-b

when an exponential expression is raised to another power, _______ the exponents

multiply

(x^a)^b =

x^ab

how do you combine bases in exponent problems, use _______ _______ to uncover common bases then combine terms

prime factorization

when multiplying two exponential expressions with different bases and the same exponents…

keep the common exponent and multiply the bases

when dividing two exponential expressions with the same base, keep the base and

subtract the exponents

when an exponential expression is raised to another power…

multiply the exponents

if the bases are not the same what can you do to try to make them the same?

prime factorization

when multiplying two exponential expressions with different bases and the same exponent, how do you solve this problem?

you keep the common exponent and multiply the bases

when dividing exponential expressions with different bases that share a common exponent, what do you do?

you keep the common exponent and divide the bases

x^a / y^a=

(x/y)^a

when a product of factors in parentheses is raised to an exponent, the exponent

applies to each factor inside the parentheses

when a nonprime base is raised to an exponent

the expression can be simplified through prime factorization and the property of exponents

when an expression contains exponents, get each base into _____ _______ and then combine

prime factorization

a√x^b =

x^b/a

fractional exponents and radicals are interchangeable, the __________ is the root’s index, and the _________ is the exponent in the radicand

denominator, numerator

how do you solve nested root problems?

separate each root, convert to fractional exponents, multiply the exponents within each root, then combine by adding the exponents

how do you solve for a variable that is the base of an exponential expression under a root?

first rewrite the radical(s) as a fractional exponent

raise both sides to the reciprocal exponent to eliminate the fractional exponents

how do you solve a comparison of roots problem?

first convert them to fractional exponents, then raise each value to the LCM of the denominators, and finally compare

principal square root

the nonnegative square root denoted by √

how do you square a binomial?

use the foil process to multiply the binomial by itself

x^-n =

1/x^n

how do you solve an equation with a quadratic expression as an exponent?

rewrite the equation so the bases are equal

apply the property: a^x=a^y then x=y

any nonzero base raised to the zero power equals

1

x^1=

x

when solving equations with like bases whose exponents are added or subtracted, first factor out the

greatest common factor of the exponents

what is the GCF of exponential terms with like bases?

the term with the smallest power

when adding n^a a total of n times, the sum is equal to

n^a+1

when adding 3^10 a total of three times, the sum is

3^10+1= 3^11

how do you solve addition/subtraction problems with exponents in the denominators?

find the LCD then apply the rule for multiplying exponential expressions to rewrite the fractions with this common denominator

when the base is greater than 1 and the exponent is an even positive integer the result is

larger

when the base is greater than 1 and the exponent is an odd positive integer, the result is

larger

when the base is greater than 1 and the exponent is a positive proper fraction, the result is

smaller

a positive number raised to an even exponent is

positive

a negative number raised to an even exponent is

positive

a negative number raised to an odd exponent is

negative

two numbers with the same base and exponents that differ by as little than 1 can be vastly different from each other, 19 to the 25th power minus 19 to the 24th power is close to

19 to the 25th power

if ten is raised to an exponent of 6, that means there are ___ zeros

6

how do you write N in scientific notation

N= a x 10^b

10^6 means the decimal moves

six places to the right

10^-4 means the decimal moves

four places to the left

to more easily compare large exponents, raise each value to the

reciprocal of the GCF of the exponents and then compare the results

how would you solve this problem, which is greater 4^12 or 6^9?

the GCF of 12 and 9 is 3 so we raise each value to the power of 1/3 and compare