Roots and Exponents

Question 1

How to simplify fraction that has a radical in denominator

Answer 1

Use the conjugate rule to simplify binomials in the form of

a + √b = a - √b
√a+b = √a-b
√a+b = √a-√b

have to multiple both numerator and denomiator by this

Question 2

Radicals raised to an even exponent

Answer 2

We already know that if n is even and x is nonnegative, ^n√x^n = |x|. always positive

for example, x^4√625 is only +5, not negative 5.

on the other hand, y^4 = 625 can be +/5

Question 3

Radicals raised to an odd exponent

Answer 3

We also know that if n is odd, ^n√x^n = x. can be negative or positive, depending on what the sign for x is.

Question 4

exponent rule

x^a * x^b =?

Answer 4

x^a+b

Question 5

exponent rule

x^a /x^b =?

Answer 5

x^a-b

Question 6

exponent rule

x^a(^b)

Answer 6

multiply a and b

x^a*b

Question 7

multiplying different bases but same exponents

Answer 7

multiple the bases but keep the exponents

2^4 * 3^4 = 6^4

Question 8

radicals converted to exponent form

Answer 8

√x = x ^1/2
3^√x=x^1/3

and in general b^(√x^a)= x^a/b

Question 9

when an exponent is under a radical square, you can..

Answer 9

half the exponent to get rid of the radical.

ex:

√8^64 = 8 ^32

Question 10

Multiple square roots

Answer 10

https://gmat.targettestprep.com/lesson/467?chapter_id=435&sidebar=true

Question 11

Number properties of unique roots

Answer 11

https://gmat.targettestprep.com/lesson/481?chapter_id=435

Question 12

Scientific notation

Answer 12

expressed in a number from 1-10. For example,

1.2 * 10^5

4.0 * 10^-3

Question 13

perfect square decimals

Answer 13

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.

A perfect decimal square must have an even number of decimal places.

Ex

√.16 = .4
√.0004 = .02

Question 14

Perfect cube roots

Answer 14

cube root of a perfect integer has exactly one third the number of zeroes to the right of the final non-zero integer

Question 15

Perfect cube decimal roots

Answer 15

The cube root of a perfect cube decimal has exactly one-third the number of decimal places as the original perfect cube.

Question 16

√x^2 will always equal

Answer 16

|x|

Question 17

if n is even for n^√x^n…

Answer 17

n^√x^n = |x| always positive

Question 18

if n is odd for n^√x^n…

Answer 18

n^√x^n = x

can be negative or positive

Question 19

Cube roots

Answer 19

The cube root of a perfect cube is an integer, while the cube root of a non-perfect cube is not an integer.

Question 20

Cube root of first 11 nonnegative perfect cubes

Answer 20

0^3 = 0
1^3=1
2^3=8
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

Question 21

common approximate square roots

Answer 21

√2 = 1.4
√3 = 1.7
√5 =2.2
√6 = 2.4
√7 = 2.6
√8 = 2.8
√10 = 3.2
√11 = 3.32

Question 22

common approximate cube roots

Answer 22

^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^√9 = 2.1

Question 23

common approximate fourth roots

Answer 23

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

Question 24

multiplying radicals

Answer 24

can only be done if they share the same index number

when multiply, smush radicals under one radical

sqrt A * sqrt B = sqrt AB