Ch. 4 Roots and Exponents Flashcards
What is √144?
12
positive ONLY (b/c the radical sign is used)
What is √x^2?
√x^2 = ∣x∣
How would you solve x^2 = 25?
x^2 = 25
√x^2 = √25
∣x∣ = 5
x = +-5
Perfect squares up to 15 x 15
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
What is a perfect square?
all prime factors have even exponents
square root is a whole number
If n is even, what is ^n √x^n ?
If n is odd, what is ^n √x^n ?
If n is even, ^n √x^n = ∣x∣
For example, ^6 √2^6 = ∣2∣ = 2 (it is NEVER -2)
If n is odd, ^n √x^n = x
(can be +ve or -ve, depending on what x is)
What is a perfect cube?
all prime factors have exponents that are multiples of 3
Cube root of a perfect cube is an integer
What are the first eleven nonnegative perfect cubes?
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000
√xy = ?
√x ⋅ √y
When can you multiply radicals?
ONLY if the index number is the same
When can you divide radicals?
ONLY if the index number is the same
When can you add/ subtract radicals?
“LIKE” radicals only = when the index AND radicand (expression under radical) is the same
(Remember PEMDAS - the radical acts as parentheses! so √(a+b) != √a+√b
What is a conjugate pair?
a pair of binomials with identical terms but parting opposite arithmetic operators in the middle of these similar terms
a binomial is an expression with two terms
examples of conjugate pairs:
a - b, a + b
a - √b, a + √b
how do you solve this?
^4 √((x + y)^4) = 16
|x + y|=2
x + y = +- 2
(absolute value is the same for binomials under a radical as single variables under a radical)
What must you ALWAYS do after you solve an equation involving a square root?
Plug the answer back in to make sure it still works
T or F: if the bases of an equation are equal, then the exponents are equal
True
a^x = a^y –> x = y
EXCEPT if a = 0 or +/-1
(x^a)(x^b) = ?
x^(a+b)
x^a / x^b = ?
x^(a-b)
(x^a)^b = ?
x^ab
(x^a)(y^a) = ?
(xy)^a
(x/y)^a =
(x^a) / (y^a)
When can you distribute an exponent?
ONLY when multiplying or dividing:
(3ab)^4 = 3^4 x a^4 x b^4
NOT when adding or subtracting:
(a+b)^4 != a^4 + b^4
If a & b are prime, what do we know about x, w, y, and z in this equation? (a^x)(b^w) = (a^y)(b^z)
x = y
w = z
This is ONLY true if a & b are NOT 0 or +/- 1
What is √√(x) = ?
(x^(1/2))^(1/2) = x^(1/4)
^a√^b√x = x^(1/ab)
if x, y and m are positive, and x > y, then can you determine x^m vs. y^m?
yes,
x^m > y^m
what is a strategy that you can use if you see a question like this:
which is larger: ^5√7 or ^4√4?
You can determine the LCM between 4 and 5 (the exponents) then raise each term by that exponent:
^5√7 = 7^(1/5)
Then you do 7^(1/5)^20 = 7^4
^4√4 = 4^(1/4)
Then you do 4^(1/4)^20 = 4^5
Now it’s much easier to calculate 7^4 > 4^5 do ^5√7 is larger.
What strategy can you use if you see a question like this:
which is larger 5^50 or 7^25?
You can find the GCF of the two exponents, then raise each term by the RECIPROCAL of the GCF.
The GCF of 50 and 25 is 25, so
5^50 ^(1/25) = 5^2
7^25 ^(1/25) = 7^1
So 5^50 is greater.
How to factor out one 50 from 50^100 = ?
= 50(50^99)
x^(-1) = ?
1/x
(x/y) ^(-z) =
(y/x) ^z
a^0 =
1
If you see addition or subtraction of like bases (2^6 - 2^4), what can you do?
Factor out the common factor!
2^4 (2^2 - 1)
2^n + 2^n = ?
When else does this rule apply?
2^(n+1)
This applies when the NUMBER of bases = the base:
2^n + 2^n = 2^(n+1)
3^n + 3^n + 3^n = 3^(n+1)
4^n + 4^n + 4^n + 4^n = 4^(n+1)
n can be ANY number:
3^9 + 3^9 + 3^9 = 3^(9+1) = 3^10
when a^x = 1, what can a & x be?
there are only three scenarios where a^x = 1:
1. a = 1
2. a = -1 & x is an even number
3. x = 0
