Roots and Exponents Flashcards
The Square Root
When the radical symbol is used, only consider the non-negative square root of the number.
The square root of a variable squared is equal to the absolute value of that variable. (x^2)(1/2) = |x|. If you get |x| = |y|, solve for two conditions. 1. X and Y are positive & equal. 2. X & Y are equal but opposite signs.
Pos/Neg Implications of Roots
If n is even, (x^n)^1/n = |x|
If n is odd, (x^n)^1/n = x
Perfect Cubes
Memorize
x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 x^3 = 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000
Estimating Non-Perfect Square Roots
Root 2 = 1.4 Root 3 = 1.7 Root 5 = 2.2 Root 6 = 2.4 Root 7 = 2.6 Root 8 = 2.8
Use the conjugate to simply radical binomials in the denominator
[a + root (b)] x [a - root (b)] = a^2-b
Solving Equations Involving Square Roots
If the solution provides you with two answers, plug them back in to the original expression (with the root) to check for extraneous solutions.
Equalities involving exponents and -1, 0, and 1
If a = 0, -1, or 1, and a^x = a^y, x may not necessarily equal y.
Exponents can be distributed over multiplication and division
(4abc)^2 = 4^2 * a^2 * b^2 * c^2
Removing/Comparing Radicals with LCM
If there are radicals multiplied across with different bases, you can isolate the desired variable by finding the LCM of the bases and raising all the radicals to the LCM to isolate the variable.
(x^5)^(1/6) * (y^3)^(1/10) = (z^2)^(1/15) –> Find the LCM, which is 30. Raise all variables to power 30.
(x^5)^(5) * (y^3)^(3) = (z^2)^2 –> now set z in terms of x and y
To compare the size of 4^(1/4) to 7^(1/5), first find the LCM of the bases and raise each expression to that base.
4^5 & 7^4 = 1,024 & 2,401 –> 7^(1/5) is larger than 4^(1/5).
Comparing Sizes of Exponents
Find the GCF of the exponents. Raise each expression to the reciprocal of the GCF. Compare the results.
5^50 vs. 7^25 –> Find GCF, which is 25
(5^50)^(1/25) vs. (7^25)^(1/25)
5^(2) vs. 7^(1)
Powers of Ten (leading and trailing zeros)
10^n will have n trailing zeros.
10^2 = 100
10^5 = 100,000
10^-n will have n-1 leading zeros
10^-2 = 1/100 = 0.01
10^-5 = 1/100,00 = 0.00001
Zeros, Decimal Places, and Perfect Squares
When a perfect square ends with an even number of zeros, the square root of such a perfect square will have exactly half the number of zeros.
If a decimal with a finite number of decimal places is a perfect square, its square root will have exactly half the number of decimal places.
Ex: (0.0004)^(1/2) “4 decimal places” = (4/10,000)^(1/2) = (2/100) = 0.02 “2 decimal places”
Cube Roots of Large and Small Perfect Cubes
The cube root of a perfect cube integer has exactly one-third the number of zeros to the right of the final nonzero digit as the original perfect cube.
Ex:
(1,000,000)^(1/3) = 100 –> 100 x 100 x 100 = 1,000,000
(27,000)^(1/3) = 30 –> 30 x 30 x 30 = 27 x 1,000 = 27,000
The cube roof of a perfect cube decimal has exactly one-third the number of decimal places as the original perfect cube.
(0.000027)^(1/3) “6 decimal places) = (27/10^6)^(1/3) = 3/10^2 = 3/100 = 0.03 “2 decimal places”
Squaring Decimals with Zeros
What is the decimal value of (0.000005)^2 = (5/1,000,000)^2 = (5/10^6)^2 = (25/10^12) = 0.000000000025.
The number of decimal places doubled. From 6 decimal places, to 12 decimal places.
Square Root of a Binomial
You can take the square root of a binomial expression, just make sure to use the absolute value.
Ex. (x+6)^2 = 49 –> |x+6| = 7 –> x + 6 = 7 and x + 6 = -7 —> x = 1, -13
