Roots and exponents Flashcards
square root of a number is always positive
square root of a squared variable is always +-
because we consider the result as an absolute value
n=square root 64
n=8
x^2=4
x=+-2
radical index
is a number found on the top left of the radical
x√2
if n is even and x non-negative
n√x^n= absolute value of x is always positive
if n is odd= x
4√10,000= 4√10^4= 10
x√y= z^x=y
by prime factorization of y we can find the result
m√z if m is even the result will be greater or equal to zero
data sufficiency question
the square root of a perfect square
is always a whole number if not perf square the result will not be a whole number
perfect square is a number whose prime factorization contains only even exponents
i.e 81=3^4
perf squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
the cube root
a number that multiplied by itself twice will produce the value under the root
i.e √125=5 =555=125
the cube root will be an integer when
it is a perfect cube- a number whose prime factorization contains only exponents that are multiple of 3
0,1,8, 27, 64, 125, 216, 343, 512, 729, 1000
also their negatives are perfect cubes
simplifying radicals
simplify the square and cube roots
approximating square roots of non-perfect squares
√2=1.4 √3=1.7 √5=2.2 √6=2.4 √7=2.6 √8=2.8
estimation of less common radicals
√70–> √81=9, √64=8
thus V70 must be between 8 and 9
since its closer to 64 than to 81 it must be closer to 8 than 9
approximating cube and fourth roots
3√2=1.3 3√3=1.4 3√4=1.6 3√5=1.7 3√6=1.8 3√7=1.9 3√9=2.1
4√2=1.2 4√3=1.3 4√4=1.4 4√5=1.5 4√6=1.6 4√7=1.6 4√8=1.7 4√9=1.7 approximate by taking the closest upper and lower root
multiplying radicals
can be multiply when the index of the square root is the same
i.e √7 x√5= √75= √35
3√25 * 3√5= 3√255= √125=5
never combine radicals with different index numbers
dividing radicals
n√a/n√b= n√a/b
multiply and divide non-radicals by non-radicals
and radicals by radicals
when expression contains both radicals and non radicals multiply the radicals (if same index) and the non-radicals
i.e 2√10*5√2= 10√20= 10√4√5=20√5
addition and subtraction root square
first we have to perform addition and subtraction under the square root before proceeding
add and subtract only like radicals
two or more radicals are like radicals if they both have the same root index and the same radicand
radicals must be removed from the denominator for the expression to be simplified
by rationalizing the denominator
i.e multiply the both the nominator and denominator by the radical in denominator
3/√5 = 3√5/√5√5= 3√5/5
conjugate in radicals
a-b its conjugate a+b
difference of squares eliminates the radical (the conjugate)
i.e 4/a-√b*a+√b/a+√b= 4a+4√b/a^2-√b^2
=4a+a√b/a^2-b
solving equations with the variable raised to an even power
the result is always positive
square root of a binomial squared
√(x+y)^2= |x+y|
exponents
represents a number of multiplications of a certain value
if the base are equal (2^x )(2^y) =
when multiplying like bases, keep the like base and add the exponent
i.e. (5^2) (5^4)= 5^6
division of like bases
when dividing like bases, keep the like base and subtract the exponents
the power to a power rule
(x^4)^2= x^8 i.e 4*2
multiply the exponents
when the exponents are same and the base are different
keep the exponent and multiply the base
(2^4)(3^4)= 6^4
x^a/y^a=
(x/y)^a
division of different bases and same exponents
keep the exponent and divide the numbers
a^x= a^y if a is not 0,1,-1
same goes for a^xb^y= a^zb^y
a fraction smaller than 1 raised to a positive exponent
will be smaller than the original fraction,
bigger the exponent smaller the fraction
exponential rule: (a/n)^-n=
(n/a)^n
