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🧮➗ > Exponent and radical SAT > Flashcards

Exponent and radical SAT Flashcards

1
Q

Which of the following expressions is equivalent to the expression above?
√75x^4/ √12x^7

A

SIMPLIFY FIRST :
√ 75 x.x.x.x : 12 x.x.x.x.x.x.x =>
√75 : 12x³

√3 . √25 : √3 . √4 x√x
5 : 2x√x

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2
Q

Which of the following expressions is equivalent to the expression above?

5¹/³ - 5 ^³/4

A

1) Radicals sum -> turns to multiplying : the big one will be ¹ . ¹ - big one ; The small one will be 1 . ^small one

  1. 5¹/³ - 5 ¹ - ³/4 . 5¹
  2. 5¹/³ - 5 ¹/³ . 5¹
    (ISOLATE) 5¹/³ ( 1 - 5)
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3
Q

Y + Y²/ Y -²/³

A

Y- ²/³ => go to the numerator

(y+y²) y²/³ => y.y²/³ + y² . y²/³
y^5/3 + y ^7 /3

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4
Q

x - 5√x + 6 = 0

A 
1) ISOLATE THE RADICAL 
x +6 = +5√x
2) Take out the radical : ² 
(x+6)² = 25x
x+12x + 36 - 25x = 0 
x - 13x + 36 = 0

sum ( -13x) and product (36) :
-9 + -4 and -9.-4

CONVERT THE SIGNALS

+ 9 e +4

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5
Q

√x² - 4x - 4 = 0

√ extends to -4x-4

A

1) TAKE OUT THE RADICAL:
(√x² - 4x - 4)² = 0²
x² - 4x - 4 = 0

sum ( -4x) and prod ( -4) = -2.-2
CONVERT THE SIGNALS = +2

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6
Q

√y² -15y - 4 = 0

ratio extends to y-15

What is the sum of all solutions

A

√y² - 15y = + 4
y² - 15y = 4²
y² - 15y - 16 = 0

-15² - 4.-16.1 = 225 + 64 = 289
- (-15) +- 17 : 2
+1 and -16

REPLACE THE ROOTS

y² - 15y = 16
1² +15 = 16 (v)
-16² - 15.16 = 16 (v)

-1 + 16 = 15

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7
Q

x - 5√x + 6 = 0

A

ISOLATE THE RADICAL:
(x+6)² = (+5√x )²
x² + 12x + 36 = 25x
x² - 13x + 36 = 0

-13² - 4.36.1
169 - 144 = 25
-(-13) +- 5 : 2
4 and 9

REPLACE THE ROOTS
4² - 13. 4 + 36 = 0 (v)
9² - 13.9 + 36 = 0 (v)

4+ 9 = 13

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8
Q

√3z + 6 - z = 2

! the √ extends to 3z+6

What is the largest solution to the above equation?

A

ISOLATE THE ROOTS
(√3z + 6)² = 2+z )²
3z + 6 = z² + 4z + 4
z² +z - 2 = 0

  • 1 + 2 = 1
  • 1.2 = -2

INVERT THE SIGNALS
-1 and +2

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9
Q

2√9x - 6 = 10 - 2 √x

A 
2√9x = 2 √x => 2.3√x = 2√x 
-16 = 8√x
8/16 = √x
2 = √x = 4
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10
Q

Don’t solve the quadratic form in traditional way, rather:

15y - 5√y = 0

A

SIMPLIFY AS POSSIBLE
:3
(3y - √y = 0) ²
9y² - y = 0

QUADRATIC = FACTOR
y (9y -1) = 0
EQUAL 0

y = 0 
9y-1= 0 
y = +1/9
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11
Q

Don1t solve the quadratic for, rather

5t = 5√t

A

factor it out and equals to 0

25t² = 25.t
25t² - 25t = 0 
5t ( 5t - 5) = 0 
5t = 0 -> t = 0 
5t-5 = 0 
5t = 5 -> 1
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12
Q

Ratio + both numerator and denominator has x
-2x + 14/3x = 4

-2x + 14 is one numerator to 3x which is denominator

A
  1. MULTIPLY the denominator in BOTH SIDES ( eliminate denominator )
    -2x+14 = 12x
    14x - 14 = 0
    14x = +14
    x = 1
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13
Q

ratio with x = ratio :

y/4y -3 = 2 /3

A 
CROSS MULTIPLY 
2y = 12y - 9 
10y - 9 = 0 
1oy = +9 
y = 9/10
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14
Q

quadratic form don’t need to be solved in trad way :

A 
FACTOR IT AND EQUALS 0 
w² = √108w )²
w² -108w = 0 
w ( w-108) = 0 
w=0 
w= +108
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15
Q

√ = the answer is only _ numbers
√100 =

a) -10
b) +10
c) -10 + 10
d) 0.1

A

b

in positive numbers

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16
Q

3/c+4 + 1/4

a) 1
b) 4/ c+ 8
c) c+16 / 4c+16

A
  1. same denominator = denominator 1. denominator 2 => MMC 4 (c+4)
  2. Multiply needed number to the same denominator

3/c+4 x 4 = 12/ 4(c+4)
+ 1/4 x 4 = 4/4(c+4)

12/ 4(c+4) + 4/4(c+4) = 16/4(c+4)
C

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17
Q

x/ x² +5x - 14 + 7/ x² +5x - 14

a) x+7/ 2x² + 10x - 28
b) x7 / x² +5x - 14
c) 1/x-2
d) 1 /x+2

A

X+7 / x² +5x - 14
NO SUCH ANSWER IN THE OPTIONS : FACTOR, SIMPLIFY

x+7 / (x - 2) (x+7)
CUT the same
c) 1/x-2

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18
Q

SIMPLIFY AS POSSIBLE
x²/x-2 + 4/2-x

x is different with 2

A

ALWAYS REMAINS THE X - number

4/2-x (.-1) => -4/x-2
x² /x-2 -4/x-2 = x² - 4 /x-2

NO SUCH ANSWER = FACTOR, SIMPLIFY
(x -2) (x+2) / (x-2) - since there is no number.x
CUT
x+2

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19
Q

5m/m² -24mn + 144m² + 2n/ m²-144n²

a) 5m² +60nm - 2n² / (m-12) (m+12)
b) 5m² + 60nm - 24n² / (m-12)² (m+12)
c) 5m² + 60nm - 24n² / (m-12) (m+12)²

A
  1. FACTOR
    5m/ (m -12n)² + 2n/ (m-12n) (m+12n) - since there is no number.x
  2. Same denominator : multiplies down and up the needed number
    5m/ (m -12n)² (. m+12n) = 5(m+12n) / (m -12n)² (m+12n)

2n/ (m-12n) (m+12n) = 2n (m-12n) / (m-12n)² (m+12n)

B

20
Q

x³+ 7x² + 10x /x² + 2x

a) x+ 5
b) x+8
c) x³ + 6x²+ 8x

A
  1. factor
    x (x² + 7x + 10) / x ( x+2)
    x ( x+5) (x+2) / x ( x+2)
  2. CUT x and (x+2)
    x+5
21
Q

if P(x) = g² + 10g + 25 and Q(x) = g+5, so P(x) . Q(x) =

a) (g³+5)²
b) (g+5)³
c) (g+5) (g+5)
d) (g²+5)²

A
  1. factor
    g²+10g+25 = (g+5)²

(g+5)² . g+5 = g+5)³

22
Q

The equation s = (t+3)²(t+2)(t+1)(t)(t-1). How many roots are there.

A

(t+3)² = -3
(t+2) (t+1) = t² + 2t + 2 -> +1
t (t-1) = t = 0 and t-1= 0 = +1

THERE ARE -3,+1,0 3 roots

23
Q

A polynomial function f is defined as f(x)=(10x-3)(4x+1)(5x-2.)
What is the sum of all of the zeros of function f?

A

10x-3 = +3/10 -> 0.3
4x + 1 = -1/4 -> -0.25
5x-2 = +2/5 -> 0.4

Sum them

24
Q

A polynomial function P is defined as P(x)=0.5(3x+5)(3x-1) parenthesis in the x‑plane?

A

0.5 ( 3x+5) (3x-1) = 0
(3x+5) (3x-1) = 0/0,5
3x+5 = 0 ; 3x-1 = 0
x = -5/3 x = +1/3

25
Q

h(t)=(t−8)¹ (t-4)² (t-2)³ (t-1)^4

The polynomial function h is defined above. How many distinct zeros does h have?

A 
each one has 1 root 
(INVERT THE SIGNALS) 
t-8 = +8 
(t-4)² = +4
(t-2)³ = +2
(t-1)^4 = +1
26
Q

(x-√3)² (x-√ 7)
Given the polynomial above, what are its zeros?

YOU ONLY NEEDS TO

A

invert the signals

  • > +√ 3
  • > +√ 7
27
Q

It is given that t+2t is a factor of the polynomial equation h=10(t^3+4t^2+t-6)
Which of the following is the graph of the equation in the t h‑plane?

A

x-intercept at -2

since is + 10 the parabola will be UPWARDS ( IN POSITIVE) WHEN GOES TO THE IFINITY

28
Q

A function s is defined as s(x)=(x-4)(x-5)²
. A function h(x)=(x−a)⋅s(x)h.
For some constant a, (x-a)³ is a factor of h. What is s(a)

A

h(x)=(x−a)⋅s(x)h -> (x-a)³ is a factor of h.
(x-a) (x-a) (x-a) -> s(x) = (x-a)²

So, factor of s(x) is (x-a)², which means that s(a)=0

29
Q

−2−x = y² - 4y +10/11
−11x+3y = 62
​ are two distinct solutions to the system of equations shown above, what is the product of the x values of the two solutions x1.x2

A

1) −2−x = y² - 4y +10/11 : +x = - ( y² - 4y +10/11) -2 ( Since there’s already a y²)

2) Replace in another equation
−11 [- ( y² - 4y +10/11) -2] +3y = 62
cut -11 with /11 and -11 x -2

y² - 4y +10 + 22 + 3y - 62 = 0
y² -y - 30 = 0

3) báskara
1² - 4.1.(-30) = 121
1 + - 11 / 2 = 6 and 5

6.5 = 30

30
Q 
y = 3-x/2
7y² = 50x - 150

are distinct solutions to the system of equations shown above, what is the product of the y1.y2

A

1) y = (3- x)/2 => x = 2y +3 ( Since there is already a y²)
2) replace it in another equation

7y² - 50 (2y+3) +150
7y² - 100y -150 + 150
7y² - 100 y

3) factor :
y ( 7y - 100) = 0 
7y-100 = 0 
y = 100/7 
y = 0

0

31
Q

y=x-2
y = x² -x -5
​

Which of the following represents all solutions (x,y)to the system of equations shown above?

a) (1,-3)
b) (-1,3)
c) (-1,-3) and ( 1,3)
d) (1,-3) and (-1,3)

A

1) equal the equations, since y=y
x-2 = x² -x -5
x² -x-x -5 + 2 = x² -2x - 3

2) -3 . 1 = -3 and -3 + 1 = -2

3) Convert the signals
! (x,y) -> the ones that has +3 and -1 in the x position

32
Q

y = -2(x-1)² + 5
3 = 2x +y
​

Which of the following is the graph of the system of equations shown above?

describe the graph

A

the parabola:
a < 0 : face down
vertex x = +1 and vertex y = +5

the linear equation : y = 3-2x
a < 0 : line decreasing

33
Q 
2x-6y = 9
7-2y= 15x - x²

what is the product of the solutions of this system of equations

A

1) eliminate y by multiplying the second equation with -3x
(.-3) 7-2y= 15x - x² (.-3)

  • 21 +6y = -45x +3x²
    2x-6y = 9
    -21 + 2x = 9 - 45x +3x²

2) Shift everything in one side
- 21 - 9 +2x +45x -3x² = 0
- 30 +47x - 3x² = 0

3) báskara
47² - 4.(-30).(-3)
2209 - 360 = 1849

-47 + - 43 / -6 = 2/3 and 15

2/3 x 15 = 10

34
Q

(m) 12 18 40 60 200
(s) 15 20 30 35 50

. The chart above shows the mass mmm of displaced snow, in tons, s seconds after the start of the avalanche. Which of the following best describes the relationship between m and s?

a) m increases exponentially by approximately 125 percent every ten seconds.
b) m] increases linearly by approximately 125 percent every ten seconds.
c) m increases exponentially by approximately 31 tons every ten seconds.
d) m increases linearly by approximately 31 tons every ten seconds.

A 
1) Indetify if is exponentially or linearly : 
18 - 12 = 6
40 - 18 = 22
60 - 40 = 20 
Linear is unbalanced. So, is exponential

2) 18/12 = 1.5
40/18 = 2.2
60/40 = 1.5
200/60 = 3.33

“every 10 seconds -> 20 and 30
40/18 = 2.2 approx. 2.25
2.25 - 1 ( 100%) = 1.25

A

35
Q 
1970 ------ ( Transitor count - in millions) 0.046
1980 ----- 0.099
1990 -------- 0.215
2000 --------0.466
2010 ---------- 1.007

The number of transistors on an average computer chip has increased exponentially over time, as shown above. Assuming this pattern will continue until the number of transistors on a computer chip reaches 10 million, in what year will that chip be produced, to the nearest 10 years?

A

number of transitors
1) find the most roundest
1980 - 0.099 = 0.01
2010 - 1.007 = 1

2010 - 1980 = 30 YEARS

( 2010 + 30 , 1.10) = (2040, 10)
10 = 10 millions

36
Q

write the equation form of exponential growth and what serve each element

A

A(t) = Ao ( 1 + r) ^t

A = final value as decimal ( after growth or decrease)
t = the time, the x-axis
Ao: the initial value
r : the constant percentage of changing rate

37
Q

An investment lost approximately 5%, percent of the balance each month for the past year. The amount of the investment on January 1th of last year was$10,000. Which of the following functions, I, models the amount of the investment (in thousands of dollars) at the end of month, n, where 1≤n≤12?

a) I(n) = 10 + 0.95n
b) I(n)=10⋅(1.05)^n
c) ,I(n)=10−0.05n
d) I(n)=10⋅(0.95) ^n

A

1) is % = exponential = have exponents : b . a ^x
2) “lost” = decreasing = a < 0 : 100% - 5% = 95%
0. 95

D

38
Q

how to identify a exponential growth rather than linear. And how to identify its answers

A

1) the graph should have a curved line - part of a parabola
2) has exponent in the answer - A(t) = Ao ( 1+r) ^t or b.a^t
3) If
is increasing = a > 1
is decreasing = a < 1
4) The rate is in %

39
Q

The present value (PV) of an investment is the amount that should be invested today at a specified interest rate in order to earn a certain amount at a future date. The amount desired is called the future value. For a future value of $10,000 which of the following functions models the present value, PV, to be invested in a savings account earning 5% percent interest compounded annually for t years?

a) PV = 10000 ( 1.05)^t
b) PV= 10000 ( 1 - 1.05)^t
c) PV = 10000 ( 1.05)^-t
d) b) PV= 10000 ( 1+ 0.05)^t

A

PV= PRESENT value ; 10000 FUTURE value; Therefore, PV should be fewer than future

C

a) (x) should be decreasing to reach PRESENT value
b) is not exponential

40
Q

The nutrition facts on a container of mini pretzels state that each serving of the mini pretzels contains 310milligrams of sodium, which is 13% percent of the recommended daily value for adults. If xxx servings of mini pretzels contain p percent of an adult’s recommended daily value of sodium, which of the following expresses p in terms of x ?

a) p= 310 ( 0.13)^x
b) p = 1.13^x
c) p = 13x
d) 310 (0.13x)

A

b(x) This exponential indicates that the last portion has less sodium that the next one.
d(x) Shows how many % has in bags of pretzels. We want to know x servings contain p percent of an ADULT

C : If 1 portion has 13 % of sodium in adult , then 2 portion would have 26% of sodium in adult

41
Q

Takada is hosting an event for 100 of her fans. For the event, she meets each of her fans one at a time at a constant rate. After 90 minutes, she has met 45% percent of her fans at the event. Which of the following equations models the number of fans, F, remaining for Takada to meet mmm minutes after the event started?

a) 100 - 0.5m
b) 100 - 0.45m
c) 100(0.45) ^m/90
d) 100(0.55) ^m/90

A

AT A CONSTANT RATE = LINEAR not exponential

A - am = per minute : 90min —— 45
1 ——- = 0.5

42
Q

When the second confidence level is + , then the estimated % is ___ and the margin error is ___

A researcher surveyed a random sample of young voters (n the U.S. and estimated that 38% percent of young voters in the U.S. intend to vote in the upcoming election. At the 95% confidence level, the margin of error for this estimation is 5%, percent. Which of the following could be the estimate at the 68% percent confidence level?

38% percent of young voters in the U.S. intend to vote in the upcoming election, with a margin of error of 7%
38%t of young voters in the U.S. intend to vote in the upcoming election, with a margin of error of 2%
27\% of young voters in the U.S. intend to vote in the upcoming election, with a margin of error of 5%
27% percent of young voters in the U.S. intend to vote in the upcoming election, with a margin of error of 2%

A

When the second confidence level is + , then the estimated % is THE SAME and the margin error is HIGHER and vice-versa

B

43
Q

Dutch elm disease is a fungal disease spread by beetles. In a city with 75000 elm trees, an arborist examined a random sample and found that 8%, percent of the elm trees she examined have symptoms of Dutch elm disease. If the margin of error is 1.2% at the 80% confidence level, which of the following best estimates, at the 80%confidence level, the number of elm trees in the city that have symptoms of Dutch elm disease?

A

margin error = 8% +1.2% and - 1.2%
75000.9,2% = 75000. 0.092 = 6900
75000 . 0.068 = 5100

44
Q

A medical school student estimates that all medical school students sleep for 5 to 8 hours each night at the 80, percent confidence level. Which of the following is a reasonable estimate of the number of hours medical school students sleep each night at the 95, percent confidence level?

6.5 to 5 hours
6 to 7 hours
3 to 10 hours
2 to 5 hours

A

5 + 8 /2 = 6.5
5 - 8 /2 = 1.5
Since the second confidant % is GREATER than the original one, the range should be GREATER too
6.5-1.5 = 5H

7 - 6 = 1h (x)
3 - 10 = 7h (yes) C

45
Q

After randomly sampling 1000 cars, a car wash determined with 95%percent confidence that the mean time needed to wash a car is 8 minutes and 56seconds with a margin of error of 7 seconds. Which of the following could be the bounds of a 90%, percent confidence interval based on the same sample?

from 8 minutes and 50 seconds to 9 minutes and 5 seconds
from 8 minutes and 40seconds to 9 minutes
from 8 minutes and 40 seconds to 9 minutes and 5 seconds
from 8 minutes and 50 seconds to 9 minutes

A

8:56 + 7 seconds - 7 seconds = 9:03 and 8:49
Since the second confidant % is LESS than the original one, the range should be less than ( 8:49 - 9:03 -> 60-49min = 11min + 03min = 14min) 14min. Furthermore, the maximum should be also less than the max of original one

D : range of 10min. And maximum 9min < 9:03

46
Q

The number of bedrooms in each house in New Mexico is recorded and shown in the bar graph to the left. The rightmost category combines all houses with 5 or more bedrooms in it. According to the bar graph, which of the following statements about the mean number of bedrooms in New Mexico houses must be true?

The mean is less than 2.67
The mean is greater than 2.67
The mean is equal to 2.67
There is not enough information to compare the mean to 2.67

A

COMBINES ALL houses with 5 OR MORE = we don’t know the exact number of houses with 5 or more bedrooms . So, it’s not equal to 2.67

If is only 5 , it’s less than 2.67 . If is more than 6, is greater than 2.67
D

47
Q

On a message board, participants listed words whose letters are all in alphabetical order, such as “almost”. The table at left lists the number of words they listed with each letter count. What was the median letter count of the words they listed?

Letter count	Number of words listed
4 letters	 12
5 letters	49
6 letters	12
7 letters	9
8 letters	1
A

12 + 49 + 12 + 9 + 1 = 83
83 /2 = 41.5 + 1 = 42.5
the median is 42

4 letters) 12 -> 5 letters) 42 + 7 = 5

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