Roots & Exponents Flashcards
What is the square root
A number that becomes the number in the square root symbol when multiplied by itself (i.e. square root of x^2 is x)
*Always positive
√4
2 (NOT -2)
Square root of negative number
Not a real number because there’s no number that can be multiplied by itself to produce a negative number
Odd root of negative number
DOES exist because a negative raised to an odd power will be negative
3√-64
-4
√x^2
|x|
√(-4^2)
= 4, if the square root and the power inside the square root is even, take the absolute value of the number
3√(-4^3)
=-4, if the square root and the power inside the square root is odd, take the actual value of the number, not the absolute value
How to know if a number is a perfect square (other than 1 or 0)
A number whose prime factorization only has even exponents
First 16 perfect squares
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225
How to know if a number is a perfect cube (other than 1 )
A number whose prime factorization has exponents that are multiples of 3
First 11 perfect cubes
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Simplifying radical expressions
Find perfect squares/cubes within the expressions (i.e. √27 = √9 x √3)
Adding/Subtracting
Can only add or subtract like radicals (i.e. 10√2 + 5√2 = 15√2)
CANNOT add different root index (square root with cube) or different radicands (the value within the radical like √4 + √9 is not √13)
√2
~1.4
√3
~1.7
√5
~2.2
√6
~2.6
√7
~2.6
√8
~2.8
Finding approximate value of radicals
- Find the perfect square/cube above it
- Find the perfect square/cube below it
- The square root/cube must be between those 2 numbers
ex. √70: 1. √81 = 9, √64 = 8, so √70 must be between 8 and 9 (closer to 8 because 70 is closer to 64 vs 81)
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√8
=2
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
Estimating roots (fourth, fifth, sixth roots)
- Find the 2 numbers that root is between
- Round closer to the value of the actual root
ex. 4√80, 2^4 = 16 and 3^4 = 81, so 4√80 must be between 2 and 3, but very close to 3 since 90 is very close to 81 (the answer is 2.99)
Multiplying radicals
Can combine like radicals only
m√a x m√b = m√ab
m√ab = m√a x m√b
*NOTE - if m is even, a and b must be non-negative
Dividing radicals
Can divide like radicals only
n√a
—— = n√a/b
n√b
*If n is even, a and b must be positive
Multiplying/dividing non-radicals and radicals
Multiple or divide the non-radicals by non-radicals and the radicals by the radicals (as long as they have the same index)
a(n√b) x c(n√d) = ac(n√bd)
a(n√b) / c(n√d) = a/c (n√b/d)
Adding/subtracting radicals
Be careful!! PEMDAS - have to do all operations within the radical before taking the root
(√a+b) DOES NOT = √a + √b
(√a-b) DOES NOT = √a - √b
GMAT rule - simplified expression
Radicals MUST be removed from the denominator for the expression to be considered simplified
Removing 1 radical from denominator
Multiply the fraction by the radical over itself
ex. x √z x√z
—– x = ——
√z √z z
Removing an expression with radical from denominatorr
Multiply the fraction by the conjugate of the denominator to clear the radical
conjugate = expression with opposite sign
a + √b (multiply by a - √b = a^2 - b) a - √b (multiply by a + √b = a^2 - b) √a + b (multiply by √a - b = a - b^2) √a - b (multiply by √a + b = a - b^2) √a + √b (multiply by √a - √b = a - b) √a - √b (multiply by √a + √b = a - b)
Conjugates of radicals in denominators
a + √b (multiply by a - √b = a^2 - b) a - √b (multiply by a + √b = a^2 - b) √a + b (multiply by √a - b = a - b^2) √a - b (multiply by √a + b = a - b^2) √a + √b (multiply by √a - √b = a - b) √a - √b (multiply by √a + √b = a - b)
Unsquaring rule
Taking the square root and including ± sign!
DONT FORGET x^2 = 4 could be +2 or -2
x^2 = 4
x = +2 or x = -2
x^2 = 0
x = 0
x^2 = -4
No real value, can’t find a real value where this would exist
Taking even root of a variable
Value could be positive or negative, MUST include ±
√(x+y)^2
= |x + y|
this is x + y = 0 AND -(x+y) = 0
Finding value of unknown when it’s inside a square root
- Isolate the square root
- Square both sides of the equation
- Plug back in - reject answers that don’t work
a^x = a^y
AS LONG AS a ≠ 0, a ≠ 1, a ≠ -1 then x =y
(a^x)(a^y) = z
x+y=z
Multiplying like bases
Keep the base and add exponents
5^2 x 5^4
=5^6
Dividing like bases
Keep the base and subtract the exponents
(xn) ^5x
- ———
(xn) ^2x
= (xn)^3x
Exponents raised to a power
Keep the base and multiply the exponents
(ab^4)^10
= (a^10)(b^40)
Equations with different bases
Reduce bases using factorization to try to find common bases, then combine
Multiplying different bases that have the same exponent
Keep the exponent and multiply the bases
(xy^z)(ab^z) =
(xyab)^z
Dividing different bases that share a common exponent
Keep the exponent and divide the bases
(1/2)^2
=(1/4)^2 = 1/16
Distributing an exponent
An exponent can ONLY be distributed over multiplication and division
( 5(a^3)(b^4)(c^8) ) ^10 =
(5^10)(a^30)(b^40)(c^80)
Raising a value to the 1/2
Square root of that number
Raising a value to the 1/3
Cube root of that number
4^1/2
=√4, =2
16^1/2
=√16, =4
27^1/3
=3√27 = 3
243^1/5
=5√243 = 3
Simplifying expressions with a series of roots
a√b√x = (x^1/b)^1/a = x^1/ab
Break out the powers, raising the right-most value to the initial fraction and multiplying by subsequent fractions
√3(√3(√3))
=3^7/8
Simplifying expressions with different roots
- Convert the root into an exponent (i.e. 60 √x = x^1/60)
- Raise each expression on either side of the equation to the LCD of BOTH exponents
- Multiply the exponent by the LCD, clear the fraction for an unknown variable and set it equal to the other variables
Which is larger 4√4 or 5√7
- Convert to fractions (4^1/4, 7^1/5)
- Raise both to the LCD (20), which converts to 4^20/4 and 7^20/5
- Simplify (4^5 = 1,024 and 7^4 = 2,401)
5√7 > 4√4
Is x>y if x^m > y^m
YES, if x, y, and m are positive, then x>y if and only if x^m > y^m
Scaling down to compare sizes of numbers
When dealing with large exponents (i.e. 5^50 vs. 7^25), you can scale numbers down to find relative value
(5^50)^1/25 vs. (7^25)^1/25
5^2 > 7^1
Simplify the expression - factoring out the GCF of an expression
Pull out the largest shared value in each term (expression separated by + or - side)
Can be variables only or can include numerical coefficients
Squaring a binomial (a + b)^2
MUST FOIL! don’t distribute the exponent
(a+b)(a+b) –> a^2 + 2ab + b^2
Raising to a negative exponent (2^-2)
- Take the reciprocal of the base and exponent and make the exponent positive (1/2^2)
- Convert if possible (=1/4)
x^-y
=1/x^y
Fraction raised to a negative exponent
- Flip the fraction and make the exponent positive
(1/4)^-3 = 4^3
(x/y)^-z (if x and y ≠ 0)
(y/x)^z
x^0
If x≠0, x^0 = 1, anything raised to the 0 power is 1
Base raised to the first power
Always itself
x^4 + x^4 + x^4 +x^4=
4x^4, CANNOT add exponents when like bases are added, only applies to multiplication
2^n + 2^n =
2^n+1
Rule applies to any base where you’re adding the same number of variables as the base
3^9 + 3^9 + 3^9 =
3^10, 3^9(1+1+1) = 3^9 + 3^1 = 3^10
Step by step: Is “x” larger than “y”
- Is the base a whole number or a proper fraction
- Is the base positive, negative, or zero
- Is the exponent even or odd
- Is the exponent positive or negative
- Is the exponent a whole number or proper fraction
If x and n are positive integers, is x^n > x
Yes, base is greater than 1 and exponent is a positive integer
i.e. when x>1 and n>1, x^n > x
If x < -1, is x^n > x
MAYBE
If x is negative, then x^n > x when n is an EVEN integer
If x is negative, then x^n < x when n is an ODD integer
If 0 < x < 1, is x^n > x
No, if the base is a positive proper fraction and exponent is a positive integer, the result is smaller and x > x^n
(i.e. 1/4 ^2 = 1/16, 1/3 ^4 = 1/81)
If -1 < x < 0, is x^n > x
YES
If the base is a negative proper fraction and exponent is EVEN, then the result is larger and x^n > x (go from negative fraction to positive fraction - i.e. -1/4 ^ 2 = 1/16, -1/3 ^ 4 = 1/81)
If base is a negative proper fraction and exponent ODD, then the result is larger and x^n > x (go from a larger negative fraction to a smaller negative fraction closer to zero - i.e. -1/4 ^ 3 = -1/64, -1/2 ^ 5 = -1/32)
If x = 0, is x^n > x
No, zero raised to any positive power is zero
0^2 = 0, 0^3 = 0, 0^n = 0 (where n >0)
Is 1^435 > 1
No, one raised to any power is one
1^2 = 1, 1^3 = 1, 1^n = 1
If x > 1 and 0 < n < 1, is x^n > x
No, if base is greater than 1 and exponent is a positive fraction, then the result is smaller x^n < x
4^1/2 = √4 = 2 27^1/3 = 3√27 = 3
If 0 < x < 1 and 0 < n < 1, is x^n > x
Yes, if base is a positive proper fraction and exponent is a positive proper fraction, the result is larger and x^n > x
(1/4) ^ 1/2 = (√1/4) = 1/2
(1/27) ^ 1/3 = (√1/27) = 1/3
x^2 < x < √x
If x is a positive fraction, this must be true - memorize
10^9
1,000,000,000, 1 zero for every power
When writing an integer that’s a power of ten as an exponent, the exponent represents the number of 0s
10^-6
=1/1,000,000
Reciprocals are powers of 10
6,000,000 in scientific notation
6 x 10^6
0.0000667 in scientific notation
6.67 x 10 ^ -5
(3.5 x 10^5)(40 x 10^6) =
= 140 x 10^11
Multiply the numbers separately from the 10’s (3.5 x 40 = 140, 10^5 x 10^6 = 10^11)
Process: Multiplying or dividing numbers in scientific notation
multiply/divide coefficients then multiply/divide powers of 10 (add or subtract exponents)
How to know if an integer ending in 0’s is a perfect square
If there are an even number of zeros to the right of the final nonzero digit. The square root will have exactly half as many trailing zeros as the perfect square
√10,000 = 100
√2,000 is NOT a perfect square (44^2 = 1,936, 45^2 = 2,025)
Process: How to know if a decimal is a perfect square
The decimal is a perfect square and there are an even number of decimal places
The square root will have half the number of decimal places
Process: how to know if a power of 10 is a cube root
If the cube root has exactly one-third of the number of zeros to the right of the final nonzero digit as the original perfect cube
3√1,000 = 10 (10x10x10 = 1,000, 1 zero vs. 3) 3√1,000,000 = 100 (2 zeros vs. 6)
Process: how to know if a terminating decimal is a cube root
If the decimal is a perfect cube and the fraction has exactly one-third of the number of decimal places as the original cube
3√0.000027 = 3/100 (0.03 x 0.03 x 0.03 = 2 decimal places vs. 6)
(0.00005)^2
0.0000000025
- Square the non-zero number (25)
- Multiply the number of decimal places by the absolute value of the exponent 5x2 = 10)
- Combine the zeros with the non-zero number to come into the total number of “zero” placeholders - 0.0000000025
