a. How were the different laws of exponents derived?
Answers
Answer:
Writing all the letters down is the key to understanding the Laws So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.Step-by-step explanation:
9514 1404 393
Explanation:
It starts with the basic idea of an exponent being a notation for the number of times the base appears as a factor.
For example, x^2 = x · x . . . . . the base x appears as a factor 2 times.
__
Then the rules for multiplication and powers of powers follow.
For example, x^2 · x^3 = (x · x) · (x · x · x) = x^5 = x^(2+3)
(x^2)^3 = (x^2) · (x^2) · (x^2) . . . . by the basic idea of repeated factors
then ...
(x^2)^3 = x^(2+2+2) = x^(2·3) = x^6 . . . . by the rule for multiplication
__
Division rules come in similar fashion.
(x^3)/(x^2) = (x · x · x)/(x · x) = x = x^(3 -2) . . . . denominator factors cancel a corresponding number of numerator factors.
Writing the reciprocal of this gives the rule for negative exponents:
(x^2)/(x^3) = (x · x)/(x · x · x) = 1/x = x^(2 -3) = x^-1 . . . . a power in the denominator is equivalent to a negative numerator power (and vice versa)
__
Rules for fractional exponents derive from the multiplication rule.
(√x)(√x) = x^1 = (x^a)(x^a) = x^(2a) . . . . . . where √x = x^a
equating versions of the exponent, we see that a=1/2, so √x = x^(1/2).
Similarly, the n-th root is x^(1/n).
__
The zero power rule comes from the division rule:
(x^2)/(x^2) = 1 = x^(2-2) = x^0 . . . . . for any non-zero x
Related Questions
z ÷ 12 = 9
PLS NEED HELP FAST
STEP BY STEP
Answers
Answer:
z = 108
z/12 = 9
12 x z/12 = 12 x 9
12 x z/12 = 12 x 9
z = 12 x 9
= 108
Answer:
z = 108
Step-by-step explanation:
z / 12 = 9
Multiply 12 and 9
z = 12 * 9
12 * 9 = 108
z = 108
Hope I helped <3
When taking a 17 question multiple choice test, where each question has 3 possible answers, it would be unusual to get or more questions correct by guessing alone?
Answers
Answer: If you’re talking on a percentage basis, it would be extremely unlikely to get a majority of the questions correct.
Step-by-step explanation:
Is 0.7 a rational or irrational number? why is a rational number
Answers
0.7 is a rational number.
Explanation:Because it is a terminating decimal expansion.
Answer:
The decimal 0.7 is a rational number. It is read as seven tenths and is equivalent to the fraction 7/10. Since it can be written as a fraction
Negative 2/5 + 1/2. Sorry for the bad photo.
Answers
Answer:
9/10
Step-by-step explanation:
Answer:
1/10
Step-by-step explanation:
Let's solve the problem,
→ -(2/5) + (1/2)
→ -(4/10) + (5/10)
→ (-4 + 5)/10
→ 1/10
Thus, the answer is 1/10.
Which answer choice shows 7.931 rounded to the nearest half?
A. 7
B. 7.5
C. 8
D. 8.5
Answers
A graphing calculator is recommended, A 60-gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 lb/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time t is given by Olt) = 18(1 - e-0.08) where t is measured in minutes and o(t) is measured in pounds. (a) How much salt is in the barrel after 5 min? (Round your answer to two decimal places.) | Iь (b) How much salt is in the barrel after 10 min? (Round your answer to two decimal places.) | ГБ lb (c) Draw a graph of the function Oſt). Qt) Qit) 20 20 15 15 10 10 5 20 t 100 40 80 60 ht 100 O 20 40 60 80 Oxt) Oxt) 20 20 15 15 10 10 5 t Lt O 20 40 60 80 100 O 20 40 60 80 100 (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as t becomes large. | | lb
Answers
t approaches infinity then salt is:
\(Q(\infty) = 18lb\)
What is volume?
Volume is a mathematical term used to describe how much three-dimensional space an object or closed surface occupies. Volume is measured in cubic units like m3, cm3, in3, etc. Volume is sometimes referred to as capacity.
The amount of salt in the barrel at time t is given by
where Q(t) is in pounds and time t is in minutes.
(a) Salt after 5 minutes i.e. t=5
\(Q(5) = 18(1 - e^{-0.08(5)} ) = 18(1 - e^{-0.4})\\\\Q(5) = 18(1 - 0.6703) = 5.93 \\\\Q(5) = 5.93lb\)
(b) Salt after 10 minutes i.e. t=10
\(Q(10) = 18(1 - e^{-0.08(10)} ) = 18(1 - e^{-0.8} )\\\\Q(10) = 18(1 - 0.4493 ) = 9.91\\\\Q(10) = 9.91lb\)
(d) When t becomes large, let say when t approaches infinity then salt is
\(Q(\infty) = 18(1 - e^{-0.08(\infty)} ) = 18(1- e^{-\infty})\\\\Q(\infty) = 18(1 - 0) = 18\\\\Q(\infty) = 18lb\)
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2π 5. Given v of magnitude 200 and direction and w of magnitude 150 and direction TT 6 find v+w. 9 2 3
Answers
The vector v has magnitude 200 and direction 120 degrees. The vector w has magnitude 150 and direction 30 degrees. The sum vector v+w has magnitude 250.1 and direction 83.5 degrees.
First, we need to convert the directions given in radians to degrees. v has a direction of 2π/3 radians, which is equivalent to 120 degrees. w has a direction of π/6 radians, which is equivalent to 30 degrees.
Next, we can break down each vector into its components using trigonometry. Let's call the x-component of v vx and the y-component of v vy. Similarly, let's call the x-component of w wx and the y-component of w wy.
For vector v
vx = 200 cos(120°) ≈ -100
vy = 200 sin(120°) ≈ 173.2
For vector w
wx = 150 cos(30°) ≈ 129.9
wy = 150 sin(30°) = 75
Now, we can add the x-components and the y-components separately to get the components of the sum vector v+w
(vx + wx, vy + wy) = (-100 + 129.9, 173.2 + 75) = (29.9, 248.2)
Finally, we can use the Pythagorean theorem and trigonometry to find the magnitude and direction of the sum vector
The magnitude of v+w is sqrt(29.9² + 248.2²) ≈ 250.1.
The direction of v+w is arctan(248.2/29.9) ≈ 83.5 degrees.
Therefore, the vector v+w has a magnitude of approximately 250.1 and a direction of approximately 83.5 degrees.
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--The given question is incomplete, the complete question is given
" Given v vector of magnitude 200 and direction 2pi/3 and w vector of magnitude 150 and direction TT/ 6 find v+w "--
Jennifer wants to purchase a book but only has $62.10 to spend. what is the maximum number of pages she can have in her book
Answers
Answer:
what is the price of for each page?
Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 9x² - 4xy + xyz (a) Find the rate of change of the potential at P(2, 4, 9) in the direction of the vector v=i+j-k. (b) In which direction does V change most rapidly at P? <56, > (c) What is the maximum rate of change at P?
Answers
The maximum rate of change at P is \($\sqrt{1732}$.\)
Given, Electrical potential \(V = 9x² - 4xy + xyz\)
The direction vector is \(v=i+j−k = < 1, 1, −1 >\)
Let P be the point \((2, 4, 9).\)
(a) The rate of change of the potential at P(2,4,9) in the direction of the vector \(v=i+j−k\) is given by the directional derivative formula, that is \($\nabla V\cdot \frac{\textbf{v}}{|\textbf{v}|}$\)
Now we find
\(∇V = $\left(\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z}\right)$\)
Therefore,
\($\frac{\partial V}{\partial x} = 18x - 4y + yz$ $\frac{\partial V}{\partial y}\)
\(= -4x + xz$ $\frac{\partial V}{\partial z} \\= xy$At P(2,4,9), $\frac{\partial V}{\partial x} = 18(2) - 4(4) + (4)(9) \\= 38$ $\frac{\partial V}{\partial y} \\= -4(2) + (2)(9) \\= 14$ $\frac{\partial V}{\partial z} \\= (2)(4) \\= 8$\)
Therefore,\($\nabla V$\) at \(P(2,4,9)\) is \($\left < 38, 14, 8\right > $\)
Now
\($\frac{\textbf{v}}{|\textbf{v}|}$ = $\frac{ < 1, 1, -1 > }{\sqrt{1^2 + 1^2 + (-1)^2}}$ \\= $ < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} > $\)
So,
\($\nabla V\cdot \frac{\textbf{v}}{|\textbf{v}|} = \left < 38, 14, 8\right > \cdot < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} > $ \\= $\frac{26}{\sqrt{3}}$\)
Therefore, the rate of change of the potential at \(P(2,4,9)\) in the direction of the vector \($\textbf{v}$ is $\frac{26}{\sqrt{3}}$.\)
(b) In which direction does V change most rapidly at P?
At P(2, 4, 9), we need to find the direction of the gradient vector\($\nabla V$.\)
We already found that \($\nabla V = < 38, 14, 8 > $.\).
The direction of maximum change is along the direction of the gradient vector \($\nabla V$ i.e, $ < 38, 14, 8 > $.\)
(c) What is the maximum rate of change at P?
The magnitude of\($\nabla V$\) at P(2,4,9) gives the maximum rate of change.
Therefore, magnitude of\($\nabla V$\)at P is \($\sqrt{38^2 + 14^2 + 8^2}$= $\sqrt{1732}$\)
Therefore, the maximum rate of change at P is \($\sqrt{1732}$.\)
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In this diagram, radius AD = 5 mm, radius BD = 12 mm and chord
CD = 8 mm. Find the exact length of AB, in surd form.
Answers
Answer:
The exact length of AB is \(3 + 4\sqrt{10}\) milimeters.
Step-by-step explanation:
Both triangles ACD and BCD are isosceles and triangles AEC, ADE, BDE and BCE are right-angled, where E is the point where line segments AB and CD meet each other. We can determine the exact length of AB by means of two horizontal right triangles (i.e. AEC, BCE) and the Pythagorean Theorem:
\(AB = \sqrt{AC^{2}-CE^{2}}+\sqrt{CB^{2}-CE^{2}}\)
If we know that \(AD = AC\), \(BC = BD\), \(AC = 5\,mm\), \(BC = 12\,mm\) and \(CE = 4\,mm\), then the exact length of AB is:
\(AB = \sqrt{(5\,mm)^{2}-(4\,mm)^{2}}+\sqrt{(12\,mm)^{2}-(4\,mm)^{2}}\)
\(AB = 3 + 4\sqrt{10}\,[mm]\)
1- Which of the following has had the greatest success treating major depression? Select one
A) MAO inhibitors
B) Lithium
C) tricyclic
D) SSRIs
2- Believing that you are being singled out for attention is a (an) Select one
A) delusion of persecution
B) delusion of reference
C) delusion of orientation
D) delusion of grandeur
Answers
1. The option D) SSRIs, The greatest success in treating major depression has been observed with the use of selective serotonin reuptake inhibitors (SSRIs).
These drugs are a type of antidepressant that work by increasing the levels of serotonin in the brain. SSRIs are considered to be the most effective medication for the treatment of depression, particularly in the long-term.
2. the option B) delusion of reference ,A belief that an individual is being singled out for attention is known as delusion of reference. Delusions are a common symptom of schizophrenia and other psychotic disorders.
A delusion of reference is characterized by the belief that everyday events or objects have a special meaning that is directed specifically towards the individual. For example, an individual with this delusion may believe that the radio is broadcasting a message meant for them or that strangers on the street are staring at them because they are part of a conspiracy.
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i need haelp on this question
Answers
Firstly, OS is the radius of the circumference. So, OS = OR = OT = OP. So:
\(OQ=TQ-OT=TQ-OS=50-16=34\)Similarly, we can find RQ by:
\(RQ=OQ-OR=OQ-OS=34-16=18\)Because QS is tangent to the circumference, the angle m\(\begin{gathered} OQ^2=OS^2+QS^2 \\ 34^2=16^2+QS^2 \\ QS^2=34^2-16^2=1156-256=900 \\ QS=\sqrt[]{900}=30 \end{gathered}\)So, OQ = 34, QS = 30 and RQ = 18.
What is the equation of the line that passes through (5,-2) and (-3, 4)?
3x+4 y-7= 0
3x-4 y-7= 0
0-3x+4 y-7 = 0
Answers
A grocery store sells a bag of 6 oranges for $4.80. If Austin spent $8.00 on oranges, how many did he buy?
Answers
4.80/6 is .80 so .80 per orange. Divide 8.00 by .80 you get 10
which of the points shown below are on the line given by the equation y=3x?
check all that apply
a. point a (1,3)
b. point b (3,1)
c. point c (3,-1)
d. point d (-1,-3)
Answers
The points (1,3) and (-1,-3) are on the line given by the given equation.
Linear Equation
An equation can be represented by a linear function. The standard form for the linear equation is: y= mx+b , for example, y=6x+7. Where:
m= the slope.
b= the constant term that represents the y-intercept.
For the given example: m=6 and b=7.
The question gives the equation y=3x from the standard form equation, you can identify that: m=3 and b=0.
For solving the question, you should check if the given coordinates comply with the equation y=3x. See below.
Coordinate (1,3) - Here x=1 and y=3, then you should replace x with 1. After that, check that the result for y is equal to the given y-coordinate.y=3*1
y=3
Therefore, the coordinate (1,3) is on the line given by the equation y=3x.
Coordinate (3,1) - Here x=3 and y=1, then you should replace x with 3. After that, check that the result for y is equal to the given y-coordinate. y=3*3 y=9Therefore, the coordinate (3,1) is not on the line given by the equation y=3x.Coordinate (3,-1) - Here x=3 and y=-1, then you should replace x with 3. After that, check that the result for y is equal to the given y-coordinate. y=3*3 y=9Therefore, the coordinate (3,-1) is not on the line given by the equation y=3x.Coordinate (-1,-3) - Here x= -1 and y=-3, then you should replace x with 3. After that, check that the result for y is equal to the given y-coordinate. y=3*(-1) y=-3Therefore, the coordinate (3,-1) is on the line given by the equation y=3x.
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Ms. Fiat gets out of her car on floor -2 and gets off at floor - 7. How far did
she travel
Answers
Answer:
She traveled 5 floors from her current location
Step-by-step explanation:
-2 - -7 = 5
The function y = f(x) is graphed below.
What is the average rate of change of the
function f(x) on the interval
-1 ≤ x ≤ 0?
Answers
Answer:
-2
Step-by-step explanation:
The explanation is in the picture
Kristen owns 61 shares of Prince Waste Collection and 45 shares of Nar Heating/Cooling. Nar Heating/Cooling is worth $7. 88 per share and Prince Waste Collection is worth $16. 59 per share. If Kristen sells all of her stocks, how much money will she receive? a. $657. 39 b. $1,366. 59 c. $1,227. 23 d. $1,296. 91.
Answers
Answer:
the answer to your question is B.) $1,366.59
Jaelyn solved the equation below and got an answer of p= 3. Is her answer correct? Show how you know or explain your thinking.
3(p+2)+9=8+6p
Answers
Answer:
No, Jaelyn is incorrect because 18 is not equivalent to 24.
Step-by-step explanation:
3(p+2)+9=8+6p
Step 1 : substitute p
3(3+2)+9=8+6x3
Step 2 : distribute 3 into 3 and 2
9+6+9=8+6x3
Step 3 : multiply 6 by 3
9 + 6 + 9 = 18
Step 4: add
24 = 18
What rule would you allow to test weather (x,y) is on line l
Answers
Just evaluate the line in the values of the point and check if the equation is true or not.
What rule would you allow to test weather (x,y) is on line l?So let's say line l has an equaiton known:
y = a*x +b
To check if a point is on the line, just replace the values of the point on the line equation, if the equation is true, then that point belongs to the line, if don't, then the point does not belong to the line.
That is the only way to check that.
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There is a hole that is 6ft deep and has a diameter of 5t how much dirt is in the hole?
Answers
I used the cylinder volume formula
Hope this helps!!!
prove that lim x→0 x^2 cos(1/x^2)=0
Answers
Therefore, according to the squeeze theorem, the limit of x^2 cos(1/x^2) as x approaches 0 is also 0: lim(x→0) x^2 cos(1/x^2) = 0.
To prove that lim(x→0) x^2 cos(1/x^2) = 0, we can use the squeeze theorem.
First, we establish the following inequalities:
-1 ≤ cos(1/x^2) ≤ 1
Since -1 ≤ cos(1/x^2) ≤ 1 for all values of x, we can multiply each side of the inequality by x^2 to obtain:
-x^2 ≤ x^2 cos(1/x^2) ≤ x^2
Now, we need to evaluate the limits of the lower and upper bounds as x approaches 0:
lim(x→0) -x^2 = 0
lim(x→0) x^2 = 0
Since both lower and upper bounds approach 0 as x approaches 0, we can conclude that the function x^2 cos(1/x^2) is "squeezed" between these two functions.
Thus, the statement is proven.
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4+3(15-2^(3)) Simplify the equation....
Answers
Answer:
25Step-by-step explanation:
\(4+3\left(15-2^3\right)\\\\3\left(15-2^3\right)=21\\\\=4+21\\\\=25\)
Answer:
25
Step-by-step explanation:
4+3(15-2^(3)
4+3(15-(2*2*2)
4+3(15-8)
4+3(7)
4+21=25
Which expression means the same as “subtract 3 from 7 and then multiply by 4”
Answers
Answer:
4 × (7 - 3)
Step-by-step explanation:
You first subtarct 3 from 7 and then multiply that answer by 4
Answer:
4 × (7 - 3)
Step-by-step explanation:
Learning task 4 perform the activity. using these electrical appliances, tell what forms of energy is/are produce when using electricity. write your answers on your answer sheet.
science
Answers
The forms of energy that the devices produce are :
(a) Light bulb gives Heat Energy .
(b) fan gives Mechanical Energy .
(c) A heater gives Thermal Energy .
The term Energy is defined as the quantitative property that is transferred to a physical system, recognizable in the performance of work and in form of heat and light.
(a) A light bulb uses electric power to produce Light and heat ,
So , the Light Bulb produces Light and Heat Energy .
(b) A Electric Fan uses the electric power to rotate the fan to give us air ,
and the rotating motor produces heat .
So , Electric Fan produces Heat and Mechanical Energy .
(c) A heater uses the electric power to produce give us heat ,
So , Heat Energy is produced by the Heater .
The given question is incomplete , the complete question is
Using the given electrical appliances, tell what forms of energy is/are produce when using electricity. write your answers on your answer sheet.
(a) A Light Bulb
(b) A Electric Fan
(c) A Heater
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If you want to put 303 items into 7 rows, how many will be in each row and how many will be left over?
Answers
Answer:
2121 Items
Step-by-step explanation:
Multiply 303 by 7 To get the answer 2121.
Answer:
43 rows & 2 left over
Step-by-step explanation:
So first, we have to do long division for 303 by 7.
303 ÷ 7 = 43 R 2
So you have 43 rows and 2 left over.
Hope this helped! :)
Write the Prime factors that 24 and 36 have in common By showing them multiplied together
Answers
The prime factors that the numbers 24 and 36 have in common are found out to be; 2 × 2 × 3
How to get prime factors?A prime factor of a number is simply factor of that number that is a prime number. This means that any of the prime numbers that can be multiplied to give the original number.
Now, we have;
Prime factors of 36; 2 × 2 × 3 × 3
Prime factors of 24; 2 × 2 × 2 × 3
The prime factors that they have in common will be; 2 × 2 × 3
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Bao bought a pair of shoes online for $46. She used a coupon code to get a 30% discount. The website also applied a 20% processing fee to the price after the discount. How much did Bao pay, in the end? Round to the nearest cent.
Answers
The amount Bao paid for the pair of shoes is given by A = $ 38.60
What is Percentage?A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, %
The difference between an exact value and an approximation to it is the approximation error in a data value. Either an absolute error or a relative error might be used to describe this error.
Percentage change is the difference between the measured value and the true value , as a percentage of the true value
Percentage change =( (| Measured Value - True Value |) / True Value ) x 100
Given data ,
Let the amount Bao paid for the pair of shoes be represented as A
Now , the initial amount for the pair of shoes be = $ 46
And , the percentage of discount = 30 %
So , the amount for the pair of shoes after discount = ( 70/100 ) x initial amount for the pair of shoes
On simplifying the equation , we get
The amount for the pair of shoes after discount = ( 70/100 ) x 46
The amount for the pair of shoes after discount = $ 32.20
Now , the percentage of processing fee = 20 %
And , the amount after the processing fees = ( 120 / 100 ) x amount for the pair of shoes after discount
On simplifying the equation , we get
The amount after the processing fees = ( 120 / 100 ) x 32.20
The amount after the processing fees = $ 38.64
Hence , the amount for the pair of shoes is $ 38.60
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the following percentages of chloride were found: 42.62%, 43.10%, 42.90%, and 42.75%. what is the average deviation from the mean
Answers
The average deviation from the mean is 0.19 that can be calculated from the mean.
An average is calculated for a set of numbers that are of the same value range.
Mean = (32.52+32.14+32.61+32.75)/4 = %32.50
Standard deviation =
|32.50-32.52| = 0.02
|32.50-32.14| = 0.36
|32.50-32.61| = 0.11
|32.50-32.75| = 0.25
(.02+.36+.11+.25)/4= 0.19
So the standard deviation is 0.19
Relative deviation = .19/32.50=0.0058 or %0.58
Yes, %32.14, because it deviates too far from the average.
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The executive of an organization is to be selected from a group
of 9 males and 4 females.
How many ways are there to choose a president, secretary and
treasurer so that two are male and one is female?
Answers
The question asks for the number of ways to choose a president, secretary, and treasurer from a group of 9 males and 4 females, with two males and one female in the positions. By using the concept of combinations, the answer is that there are 144 ways to make this selection.
To select a president, secretary, and treasurer from a group of 9 males and 4 females, such that two are male and one is female, we need to determine the number of possible combinations.
To begin, we will select 2 males out of 9. This can be done in C(9,2) ways, which is calculated using the combination formula:
C(n,r) = n! / (r!(n-r)!)
where n is the total number of options and r is the number of options to be selected. In this case, n = 9 and r = 2. Therefore,
C(9,2) = 9! / (2!(9-2)!) = 9! / (2!7!) = (9 * 8) / (2 * 1) = 36.
Next, we need to select 1 female out of 4. This can be done in C(4,1) ways, which is:
C(4,1) = 4! / (1!(4-1)!) = 4! / (1!3!) = 4.
Finally, we multiply these two results together to obtain the total number of ways to select the president, secretary, and treasurer:
36 * 4 = 144.
Therefore, there are 144 ways to choose a president, secretary, and treasurer such that two are male and one is female.
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