Ch. 4 Roots and Exponents

Question 1

What is √144?

Answer 1

12
positive ONLY (b/c the radical sign is used)

Question 2

What is √x^2?

Answer 2

√x^2 = ∣x∣

Question 3

How would you solve x^2 = 25?

Answer 3

x^2 = 25
√x^2 = √25
∣x∣ = 5
x = +-5

Question 4

Perfect squares up to 15 x 15

Answer 4

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Question 5

What is a perfect square?

Answer 5

all prime factors have even exponents
square root is a whole number

Question 6

If n is even, what is ^n √x^n ?
If n is odd, what is ^n √x^n ?

Answer 6

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)

Question 7

What is a perfect cube?

Answer 7

all prime factors have exponents that are multiples of 3
Cube root of a perfect cube is an integer

Question 8

What are the first eleven nonnegative perfect cubes?

Answer 8

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000

Question 9

√xy = ?

Answer 9

√x ⋅ √y

Question 10

When can you multiply radicals?

Answer 10

ONLY if the index number is the same

Question 11

When can you divide radicals?

Answer 11

ONLY if the index number is the same

Question 12

When can you add/ subtract radicals?

Answer 12

“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

Question 13

What is a conjugate pair?

Answer 13

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

Question 14

how do you solve this?
^4 √((x + y)^4) = 16

Answer 14

|x + y|=2
x + y = +- 2

(absolute value is the same for binomials under a radical as single variables under a radical)

Question 15

What must you ALWAYS do after you solve an equation involving a square root?

Answer 15

Plug the answer back in to make sure it still works

Question 16

T or F: if the bases of an equation are equal, then the exponents are equal

Answer 16

True
a^x = a^y –> x = y
EXCEPT if a = 0 or +/-1

Question 17

(x^a)(x^b) = ?

Answer 17

x^(a+b)

Question 18

x^a / x^b = ?

Answer 18

x^(a-b)

Question 19

(x^a)^b = ?

Answer 19

x^ab

Question 20

(x^a)(y^a) = ?

Answer 20

(xy)^a

Question 21

(x/y)^a =

Answer 21

(x^a) / (y^a)

Question 22

When can you distribute an exponent?

Answer 22

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

Question 23

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)

Answer 23

x = y
w = z

This is ONLY true if a & b are NOT 0 or +/- 1

Question 24

What is √√(x) = ?

Answer 24

(x^(1/2))^(1/2) = x^(1/4)

^a√^b√x = x^(1/ab)

Question 25

if x, y and m are positive, and x > y, then can you determine x^m vs. y^m?

Answer 25

yes,
x^m > y^m

Question 26

what is a strategy that you can use if you see a question like this:
which is larger: ^5√7 or ^4√4?

Answer 26

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.

Question 27

What strategy can you use if you see a question like this:
which is larger 5^50 or 7^25?

Answer 27

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.

Question 28

How to factor out one 50 from 50^100 = ?

Answer 28

= 50(50^99)

Question 29

x^(-1) = ?

Answer 29

1/x

Question 30

(x/y) ^(-z) =

Answer 30

(y/x) ^z

Question 31

a^0 =

Answer 31

1

Question 32

If you see addition or subtraction of like bases (2^6 - 2^4), what can you do?

Answer 32

Factor out the common factor!
2^4 (2^2 - 1)

Question 33

2^n + 2^n = ?
When else does this rule apply?

Answer 33

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

Question 34

when a^x = 1, what can a & x be?

Answer 34

there are only three scenarios where a^x = 1:
1. a = 1
2. a = -1 & x is an even number
3. x = 0