Roots and exponents

Question 1

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

Answer 1

n=square root 64
n=8
x^2=4
x=+-2

Question 2

radical index

Answer 2

is a number found on the top left of the radical

x√2

Question 3

if n is even and x non-negative
n√x^n= absolute value of x is always positive
if n is odd= x

Answer 3

4√10,000= 4√10^4= 10

Question 4

x√y= z^x=y

Answer 4

by prime factorization of y we can find the result

Question 5

m√z if m is even the result will be greater or equal to zero

Answer 5

data sufficiency question

Question 6

the square root of a perfect square

Answer 6

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

Question 7

the cube root

Answer 7

a number that multiplied by itself twice will produce the value under the root
i.e √125=5 =555=125

Question 8

the cube root will be an integer when

Answer 8

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

Question 9

simplifying radicals

Answer 9

simplify the square and cube roots

Question 10

approximating square roots of non-perfect squares

Answer 10

√2=1.4 
√3=1.7
√5=2.2
√6=2.4
√7=2.6
√8=2.8

Question 11

estimation of less common radicals

Answer 11

√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

Question 12

approximating cube and fourth roots

Answer 12

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

Question 13

multiplying radicals

Answer 13

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

Question 14

dividing radicals

Answer 14

n√a/n√b= n√a/b

Question 15

multiply and divide non-radicals by non-radicals

and radicals by radicals

Answer 15

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

Question 16

addition and subtraction root square

Answer 16

first we have to perform addition and subtraction under the square root before proceeding

Question 17

add and subtract only like radicals

Answer 17

two or more radicals are like radicals if they both have the same root index and the same radicand

Question 18

radicals must be removed from the denominator for the expression to be simplified

Answer 18

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

Question 19

conjugate in radicals

Answer 19

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

Question 20

solving equations with the variable raised to an even power

Answer 20

the result is always positive

Question 21

square root of a binomial squared

Answer 21

√(x+y)^2= |x+y|

Question 22

exponents

Answer 22

represents a number of multiplications of a certain value

Question 23

if the base are equal (2^x )(2^y) =

Answer 23

when multiplying like bases, keep the like base and add the exponent
i.e. (5^2) (5^4)= 5^6

Question 24

division of like bases

Answer 24

when dividing like bases, keep the like base and subtract the exponents

Question 25

the power to a power rule

Answer 25

(x^4)^2= x^8 i.e 4*2

multiply the exponents

Question 26

when the exponents are same and the base are different

Answer 26

keep the exponent and multiply the base

(2^4)(3^4)= 6^4

Question 27

x^a/y^a=

Answer 27

(x/y)^a

Question 28

division of different bases and same exponents

Answer 28

keep the exponent and divide the numbers

Question 29

a^x= a^y if a is not 0,1,-1

Answer 29

same goes for a^xb^y= a^zb^y

Question 30

a fraction smaller than 1 raised to a positive exponent

Answer 30

will be smaller than the original fraction,

bigger the exponent smaller the fraction

Question 31

exponential rule: (a/n)^-n=

Answer 31

(n/a)^n