Module 5: Roots and Exponents Flashcards
____s and ____s only have one value: ______
Square roots and all radicals with even indexes (sqrt4, 6, etc)
Always only have the NON-NEGATIVE (or positive) value
The square root of a negative number is ______, because ______
not a real number
because there is no number that can be multiplied by itself to produce a negative number
13^2
169
sqrt(169)
13
14^2
196
sqrt(196)
14
15^2
225
sqrt(225)
15
cube root of 27
3
cube root of 64
4
cube root of 125
5
cube root of 216
6
cube root of 343
7
cube root of 512
8
cube root of 729
9
cube root of 1,000
10
3^3
27
4^3
64
5^3
125
6^3
216
7^3
343
8^3
512
9^3
729
10^3
1,000
Simplifying Radicals (roots)
For non-perfect-squares under the square root sign: break the number into factors to try to find perfect squares within the number. Execute the square root function on those numbers and pull them out of the radical. Leave whatever is left inside the radical. This is the most simplified form.
sqrt(50)
sqrt(25 x 2)
5(sqrt(2))
sqrt(2)
~1.4
sqrt(3)
~1.7
sqrt(5)
~2.2
sqrt(6)
~2.4
sqrt(7)
~2.6
sqrt(8)
~2.8
Estimate the value of large, uncommon radicals
Calculate the nearest perfect square above & below the given radical; estimate a value based on its relation to those.
sqrt(70)
Less than sqrt(81) (9); greater than sqrt(64) (8)
A little closer to 64 than 81
~8.4
Employ this same strategy for cube roots, fourth roots, etc.
2.8 =
sqrt(8)
2.6 =
sqrt(7)
2.4 =
sqrt(6)
2.2 =
sqrt(5)
1.7 =
sqrt(3)
1.4 =
sqrt(2)
Dividing & Multiplying Radicals
Radicals can be multiplied and expressed as one radical if they have the same index number (squares & squares; cubes & cubes; NOT squares & cubes.)
sqrt(7) * sqrt(5) = sqrt(35)
Same logic applies to division.
sqrt(54) / sqrt(6) = sqrt(54/6) = sqrt(9) = 3
n-root(x^n) =
If N is even: |x|
If N is odd: x
Adding & Subtracting Radicals
Due to the order of operations:
sqrt(a + b) != sqrt(a) + sqrt(b)
sqrt(a - b) != sqrt(a) - sqrt(b)
Like radicals can’t be combined for the same result, as with division or multiplication.
If there is addition/subtraction under the same radical sign, it should be performed before doing the radical.
Two radicals can only be added or subtracted if ___
they are LIKE RADICALS:
1) same root index (square root, cube root, etc)
AND
2) the same radicand (expression under the radical.)
Fractions with radicals must be simplified by doing this:
Removing all radicals in the denominator of the fraction
When there is a single-term radical (polynomial) in a denominator of a fraction, I will ____
simplify by Rationalizing the Denominator: Remove the radical by multiplying the fraction by the radical over itself.
x / sqrt(a)
[x / sqrt(a)] * [sqrt(a) / sqrt(a)] = [x * sqrt(a)] / a
^ this is the correct, simplified form
Binomial
An expression with two terms that are either added or subtracted:
a - b
or
a + sqrt(b)
Conjugate pairs & important result
Binomial pairs with flipped signs. Nothing else changes between the two.
Conjugate of a + b: a - b
Because (a + b) (a - b) = a^2 - b^2: Product of conjugate pairs equals the difference of squares.
When the denominator of a fraction contains a binomial with at least one radical, I will ____
simplify by multiplying the fraction by the conjugate of the denominator divided by itself.
2 / [a + sqrt(b)]
Multiply by: [a - sqrt(b)] / [a - sqrt(b)]
Simplified form:
a^2 - b
The square root of a number is always ____
positive. Furthermore, a negative number as a radicand is impossible.
Solving Equations with Square Roots: Rules (2)
1) If the variable we’re solving for is under a radical, isolating the radical must be the first step.
2) Always TEST the equation/result to make sure it’s correct
(x^a)(x^b) =
x^(a + b)
If x != 0, x^a/x^b =
x(a - b)
Power To A Power rule
(x^a)^b = x^ab
When a power is raised to a power, you can multiply those exponents.
Solving Equations with Exponents: Most important rule
1) If the bases are not the same, attempt to make them the same so that you can then drop them and work only with the exponents.
How to Multiply Different Bases & Like Exponents
Keep the common exponent and multiply the bases.
2^3 * 3^3 = 6^3
How to Divide Different Bases & Like Exponents
Keep the common exponent and divide the bases.
12^4 / 3^4 = 4^4
When a parenthetical has an exponent, the exponent applies to ___
every item in the parenthetical that is being multiplied and/or divided. (otherwise it only applies to the result after multiplication/division)
Prime Factorization with Exponents.
Put the base in parentheses, prime factorize it, then apply the exponent to each element in the parentheses.
6^80 = 3^80 x 2^80
Fractional exponents
can be rewritten as roots.
The denominator of the fraction = the index of the radical.
Numerator of the fraction = raise the values inside the radical to that power (usually 1)
Multiple Square Roots: How to simplify
Translate the roots to exponential fractions, and multiply them to solve.
sqrt(sqrt(3)) = 3^(1/2) * 3^(1/2) = 3^(1/4)
[(x^1/b)]^1/a = x^(1/b * 1/a) = x^(1/ab)
When trying to simplify an equation with radicals with different indices, I will
simplify by raising both sides of the equation to the Lowest Common Denominator (LCD) of the indices. (convert the roots to fractional exponents to make it easier to visualize.)
Comparing Size of Numbers: High Index Radicals
1) Find the LCD of the radical indices.
2) Multiply each index by the radical; this should simplify the exponent so it’s easy enough to calculate and compare the sizes straight up.
Example:
3root(3)
4root(5)
LCD of 3 & 4 = 12
[3^(1/3)]^12 = 3^4 = 81 [5^(1/4)]^12 = 5^3 = 125
4root(5) is bigger.
Comparing Size of Numbers: Large Positive Exponents
1) Find the GCF of the exponents.
2) Raise the exponent to the reciprocal of the GCF; this should simplify the exponents so it’s easier to calculate and compare sizes straight up.
Example:
5^50, 7^25
GCF of 25 & 50: 25
[5^50]^(1/25) = 5^2 = 25 [7^25]^(1/25) = 7^1 = 7
5^50 is bigger
Factoring Exponential Notation: An example
x(x^9) = (x)(x)(x)(x^7) = x^3 * x^7
Move notation around when possible to make things easier.
1) What is the GCF of (4x^2)(y^4) + (2x^3)(y^2)?
2) How can we use this information?
1) GCF = 2x^2 * y^2
2) Factor it out to help with simplification.
(2x^2)(y^4) + (x^3)(y^2) = x^2 * y^2 (y^2 + 2x)
How to Square a Binomial
Recall that binomials are the ADDITION or SUBTRACTION of two terms. Must be handled differently than multiplication/division.
(ab)^2 = (a^2b^2)
(a + b)^2 NOT EQUAL TO (a^2 + b^2)
Must multiply the binomial by itself and use FOIL.
Remember the common ones that were memorized in previous chapters.
(a + b)^2 = (a + b)(a + b) = a^2 + b^2 + 2ab
How to Deal with Negative Exponents: 2 steps & a note
1) Take the reciprocal of the base. Keep the exponent in the denominator of the new reciprocal (or numerator, if it started in the denominator.)
2) Make the exponent positive.
4^-2 = (1) / [4^(1/2)] = 1/2
Note: Remember that this rule works BACKWARDS AND FORWARDS!
1 / 4^-3 = 4^3 = 64
Addition or Subtraction of Like Bases or Like Radicals
You CAN’T add exponents/radicals when adding or subtracting; that’s only for multiplication or addition.
You CAN factor out the GCF of the terms.
2^10 + 2^11 + 2^12
2^10 (1 + 2^1 + 2^2)
2^10 * 7
Adding Like Bases with Equal Exponents - a trick
2^4 + 2^4 = 2^5
3^4 + 3^4 + 3^4 = 3^5
4^4 + 4^4 + 4^4 + 4^4 = 4^5
y^n summated y times = y^(n+1)
You can just add one to the exponent.
x^2 < x < sqrt(x) if _____
If x is between 0 and 1, then x^2 < x < sqrt(x).
If the base of a term is less than -1 and the exponent is an even positive integer, then the result is _____ than the base.
Example: (-4)^2
larger
If the base of a term is less than -1 and the exponent is an odd positive integer, then the result is _____ than the base.
Example: (-2)^5
smaller
If the base is a positive proper fraction and the exponent is an even positive integer, the result is ____ than the base.
Example: (1/2)^2
smaller
If the base is a negative proper fraction and the exponent is an even positive integer, the result is ____ than the base.
Example: (-1/4)^2
larger
If the base is a negative proper fraction and the exponent is an odd positive integer, the result is ____ than the base.
Example: (-1/4)^3
larger
(-1/4)^3 = -1/64
If the base is a positive proper fraction and the exponent is an odd positive integer, the result is ____ than the base.
Example: (1/4)^3
smaller
If the base is greater than 1 and the exponent is a positive proper fraction, the result is _____ than the base.
Example: 4^(1/2)
smaller
If the base is a positive proper fraction (less than 1) and the exponent is a positive proper fraction, the result is _____ than the base.
Example: (1/4)^(1/2)
larger
(1/4)^(1/2) = sqrt(1/4) = 1/2
When I see a PS question with the words “closest to” or similar, I will (+ strategy)
take that as a cue to estimate.
Convert numbers to the closest 10^x value. This should help estimate the # of digits
In powers of 10 - (example: 10^3) - the exponent represents _______
the number of zeroes to the right of 1 in the expanded form of the number.
NOT the number of digits in the number.
10^3 = 1000
In powers of ten with a negative exponent - (example: 10^-3) - the absolute value of the exponent represents ________
the number of zeroes that exist to the right of the 1 in the denominator of the fraction.
10^-3 = 1 / 1,000
(note: NOT the number of preceding zeroes in a decimal form. That is n-1, where n is the absolute value of the exponent.)
Scientific Notation & 3 components
1) Coefficient (must be a single digit, 1-9; can have additional digits after a decimal point)
2) Base of 10
3) Exponent to which the base is raised. This indicates the number of decimal places to which we need to move the decimal place of the coefficient to get the real number. (negative exponent = move it left, positive = move it right)
Example: 93,000,000 = 9.3 x 10^7
Multiplication and division with scientific notation (or near scientific notation) - 3 steps
1) Fix the form to be in scientific notation, if necessary.
(3.5 x 10^5) * (40 x 10^6)
40 is not a valid coefficient because it’s greater than 10; adjust the decimal & exponent accordingly
(3.5 x 10^5) * (4 x 10^7)
2) Multiply the two coefficients and the two power of ten/exponent combos.
(3.5 * 4) = 14
(10^5) * (10^7) = 10^12
14 x 10^12
3) If necessary, adjust the coefficient to be between 1 and 9 again (to match correct scientific notation.) NOTE: this may not be required in the question, so pay attention.
14 x 10^12 –> 1.4 x 10^13
Finding Perfect Squares & Cubes based on Trailing Zeroes (3 points to know)
- If a number has an even number of zeroes to the right of its final nonzero digit, it may be a perfect square.
- The square root of the perfect square will have half as many trailing zeroes.
- For cubes, the cube root has one third as many trailing zeroes. (So perfect cubes with zeroes must have a number of trailing zeroes that is divisible by 3.)
Squaring Decimals
When decimals are squared, the number of digits in the decimal will double. (Same logic for cubes: digits will triple.
Example:
(0.06)^2 = 0.0036 (from 2 to 4 total digits, ending with 36, the square of 6)
(0.06)^3 = 0.000216 (from 2 to 6 total digits, ending with 216, the cube of 6)
