Roots and Exponents

Question 1

sqrt(x^2) =

Answer 1

|x|

Question 2

for even n, (x^n)^(1/n) =

Answer 2

|x| (always positive)

Question 3

for odd n, (x^n)^(1/n) =

Answer 3

x (can be positive of negative)

Question 4

sqrt(2) =

Answer 4

1.4

Question 5

sqrt(3) =

Answer 5

1.7

Question 6

sqrt(5) =

Answer 6

2.2

Question 7

sqrt(6) =

Answer 7

2.4

Question 8

sqrt(7) =

Answer 8

2.6

Question 9

sqrt(8) =

Answer 9

2.8

Question 10

radicals can be combined via multiplication if they the same..

Answer 10

index number ie. sqrt(5)*sqrt(2)=sqrt(10) since index of radical is 2

Question 11

can radicals with different index numbers be combined?

Answer 11

no

Question 12

what is the radicand

Answer 12

the expression under the radical

Question 13

radicals can be added and subtracted if….

Answer 13

they are “like” (same root index and radicand)

Question 14

Given the binomial a-sqrt(b), what is its conjugate

Answer 14

a+sqrt(b)

Question 15

what is the product of a conjugate pair?

Answer 15

difference of squares
(a-b)(a+b)=a^2-b^2

Question 16

T/F: (x/y)^(-n) = (y/x)^n

Answer 16

True

Question 17

is it always necessary to check solutions of equations with radicals?

Answer 17

YES

Question 18

In what cases does a^x=a^y not nessarily imply that x=y?`

Answer 18

if a=0,-1, 1

Question 19

x^a * x^b = ?

Answer 19

x^(a+b)

Question 20

For x != 0, (x^a)/(x^b) = ?

Answer 20

x^(a-b)

Question 21

(x^a)^b =

Answer 21

x^ab (multiply exponents)

Question 22

for different bases and like exponents
(x^a)(y^a) =

Answer 22

(xy)^a *almost acts like distributive property

Question 23

*for different bases and like exponents
(x^a)/(y^a) =

Answer 23

(x/y)^a

Question 24

Exponents can be distributed across (multiplication & division) or (addition & subtraction)?

Answer 24

multiplication & division

Question 25

what is a convenient way to compare sizes of radicals that are practically impossible to compute?

Answer 25

raise each radical expression to the LCD of the root indices, and compare the sizes of the resulting expressions (which will just be exponent expressions). This works because of the following: for x, y, and m >0, then x>y iff x^m > y^m
Ex: 2^6 vs 3^4 — raise each side to (1/2) since 2 is the gcf of 4,6. Expressions reduce to 2^3 vs 3^2, which is easy to evaluate.. 8<9, and those 2^6 (which is 64) < 3^4 (which is 81)

Question 26

2^(-2) can be re-expressed as?

Answer 26

1/(2^2)

Question 27

When multiplying or dividing two numbers in scientific notation, can the coefficients and power of 10 be dealt with separately, and then combined at the end?

Answer 27

Yes, for example:
3.510^5 x 4010^6.. first multiply 3.540 to get 140, then tack on the product of the powers of 10 for a final product of 14010^11 (1.4*10^13 if answer needed in scientific notation)

Question 28

sqrt(10,000) = ?
sqrt(81,000,000) = ?

Answer 28

sqrt(10,000) = 100
sqrt(81,000,000) = 9,000

When a perfect square ends with an even number of zeros, the square root of that perfect square will have exactly half as many trailing zeros as the original number

Question 29

sqrt(0.0004) = ?
sqrt(0.000081) = ?

Answer 29

sqrt(0.0004) = 0.02
sqrt(0.000081) = 0.009

For a terminating decimal that is a perfect square, its square root will have exactly half as many digits as the original number (not half as many zeros!), thus a perfect square decimal has an even number of digits

Question 30

cuberoot(27,000) =?
cuberoot(1,000) = ?

Answer 30

cuberoot(27,000) = 30
cuberoot(1,000) = 10

Since cube root of a perfect cube will have exactly 1/3 the number of zeros as the perfect cube

Question 31

cuberoot(0.000027) = ?

Answer 31

cuberoot(0.000027) = 0.03

Since cube root of a perfect cube decimal will have exactly 1/3 the total number of decimal places (not just zeros decimal places!) as the original perfect cube decimal

Question 32

(0.005)^2 =

Answer 32

0.000025
*total of 6 digits b/c of three initial -> 6 (double) total, then 5^2 = 25 (digits) and fill in remaining four as zeros

Protocol for squaring decimals is that the squared quantity will have a total of twice the number of existing decimal places, comprised of the square of the nonzero at the end with zeros as the remaining digits

Question 33

(0.06)^3 = ?

Answer 33

(0.06)^3 = 0.000216
b/c 6^3 = 216, and b/c since 0.06 has two decimal places, (0.06)^3 will have 2*3 decimal places total

Question 34

T/F: 2^n + 2^n = 2^(n+1)?

Answer 34

true - because of factoring