Roots and Exponent Flashcards
Square root -> never a negative number
sqrt(100)= 10 and never -10
sqrt(-16)
not a real number
-> we can’t find the real number with negative sqrt unlike positive
cuberoot(-64)
can be a real number (-4-4-4)
-> we can find positive number for negative cubic root
nsqrt(x^n)
|x| -> always positive
Square root of a whole number
whole number
Square root of non-perfect square
not a whole number
1st 11 cubic roots
1) 0^3= 0
2) 1^3=1
3) 2^3= 8
4) 3^3= 27
5) 4^3= 64
6) 5^3= 125
7) 6^3= 216
8) 7^3= 343
9) 8^3= 512
10) 9^3= 729
11) 10^3= 1000
cubicroot(-8)
-2-2-2= -8
When simplying radicals with square roots
simplify any perfect square and cubes
Approximate square roots
try to find the root that is inbetween and estimate it
cuberoot(2)
1.3
cuberoot(3)
1.4
cuberoot(4)
1.6
cuberoot(5)
1.7
cuberoot(6)
1.8
cuberoot(7)
1.9
cuberoot(8)
2
cuberoot(9)
2.1
fourthroot(2)
1.2
fourthroot(3)
1.3
fourthroot(4)
1.4
fourthroot(5)
1.5
fourthroot(6)
1.6
fourthroot(7)
1.6
fourthroot(8)
1.7
fourthroot(9)
1.7
sqrt(5)*sqrt(7) can be written as
sqrt(5*7)
-> to multiply them need to have same root
cubic(25)*cubic(5) can be expressed as
cubic(25*5)
If they don’t have the same root, can you combine them?
No you can’t
-> sqrt(100)*cubic(27)-> you can’t just multiply 100 and 27 n
Dividing radicals can be combined
- > sqrt(54)/sqrt(6)= sqrt(54/6)
- > cubic(24)/cubic(3)= cubic(24/3)
When expression has radicals and nonradicals
multiply radicals with radicals and nonradicals with nonradicals
-> 2sqrt(10)4sqrt(7)= 24sqrt(10*7)
You can add and subtract radicals that
are like: if they have the same root index and the same radical expression under the radical
-> 9sqrt(4)+2sqrt(4)= 11sqrt(4)
Remove the radicals in the denominator by
multiplying the denomiantor by the sqrt
Conjugate when you have radicals and non radicals being added/subtracted
(a-sqrt(b)) +> conjugate (a+sqrt(b))
Square root is always
postive in expection of 0
x^2
|x|
- > x^2=4
- > x=+-sqrt(4)
- > = +-2
even indexed root is
always postive
sqrt((x+y)^2)
|x+y|
when we have square root in an equation
isolate the square root and solve it
- > 3sqrt(x-1)-7= 2
- > 3sqrt(x-1)= 9
If n is a postive integer, m^n means
n factors m being multiplied
- > n^5= nnnnn -> not same as (5n)
- > 2^4= 22222
- > (xy)^3= (xy)(xy)(xy)
If a^x=a^y
x=y
a^x* a^y = a^z
x+y=z
When multiplying like base
add the exponent
-> (x^a)(x^b)= x^(a+b)
Dividing like bases
subtract the exponents
- > x^a/x^b= x^(a-b)
- > (xn)^5x/(xn)^2x
When there are exponent that are being raised to a power
multiply thos exponents
- > (x^a)^b= x^ab
- > (x^4)^2= x^8
If the bases are not same
try to make the bases same and add them
Mulitplication of different bases and like expoent
multiply the base
- > (2^4)(3^4)= 6^4
- > (x^a)(y^a)= (xy)^a
Division of different bases and like exponent
keep the common exponent and divide th bases
- > x^a/y^a= (x/y)^a
- > (12^4)/(3^4)= (12/3)^4
When non-prime factorization raised to the power
can be reduced through prime factorization
-> 6^80= (3*2)^80
Can simplify the fractions and multiplication with
prime factorization
radicals can be expressed as exponents
sqrt(x)= x^1/2
Exponents notation can be factored
-> you can factor the GCF out
x(x^9)
Raising a base to a negative exponent by
take the reciprocal of the base and exponent while turning the negative exponent into a postive exponent
- > 2^-2= 1/(2^2)
- > 10^-4 = 1/10^-4
- > x^-y= 1/x^y
(x/y)^-z
(y/x)^z
non zero base raise to 0 equals (x^0)
1
base raise to the value of the expression (5^1)
is simply the base
-> 5^1= 5
2^n+2^n=
2^n+1
when we need to add or subtract
use LCM like you sue it for others
If we need to compare the fractions and have exponent in the denominator
you can find LCM
Base is less than -1 and exponent is an even positive integer
- result is larger
- > (-4)^2= 16
Base is less than -1 and exponent is an odd positive integer
- result is smaller
- > (-4)^3= 64
Base is positive proper fraction and exponent is even positive integer
- > Result is smaller
- > (1/4)^2= 1/16
Base is negative proper fraction and exponent is even positive integer
(-1/4)^2= 1/16
Base is positive proper fraction and exponent is odd positive integer
- > result is smaller
- > (1/4)^3= 1/64
Base is negative proper fraction and exponent is odd positive integer
- > result is larger
- > (-1/4)^3= -1/64
0
x^2
Two number with same base and exponent can differ
by as little as 1
- > 9^7= 4,782,969
- > 9^8= 43,046,721
power of ten raised to exponent
just add a 0
to the 1
-> 10^1= 10
-> 10^2= 100
can be written in scientific notation
-> 6*10^-6
= 6,000,000
-> to convert into scientific notation, coefficient must be between 1 and 10
Converting to the scientific notation: n is greater than 10
move the decimal point enough to the left until it becomes a number between 1 and 10
-> 9.3*10^7= 93,000,000
If n is between 0 and 1
Move the decimal point enough places to the right until it becomes a number between 1 and 10
-> 0.0000678= 6.78*10^-5
When multiplying or dividing number in scientific notation or number almost in scientific notation
the coefficient can be multiplies or divided separately from the power of ten
When a perfect square root ends with an even number of zeros
square roots of such a perfect square will have exactly half the number of zeroes to the right of the final nonzero digit as a perfect square
- > sqrt(10,000)= 100
- > sqrt(81,000,000)= 9000
If a decimal with a finite number of decimal places is a perfect square, its square root will have
exactly half the number of decimals places
-> thus a perfect square decimal must have an even number of decimals
The cube root of a perfect cube integer has
exactly one third the number of zeroes
-> cuberoot(1,000)= 10
Squaring decimal with zeroes
i.e (0.000005)^2 -> we have 6 decimals
-> 2*6= 12
-> as 5^2 has 2 decimals we add 10 zeroes
-> (0.000005)^2
= 0.000000000025
