Which expressions are equivalent to the one below? Check all that apply.16xA.4• 42xB.42 • 4xC.42xD.4x • 4xE.(4 • 4)xF.4 • 4x
Answers
Explanation
In the question we have
[tex]4^{2x}=(4^2)^x=16^x[/tex]Also
[tex](4\times4)^x=16^x[/tex]And
[tex](4^^x\times4^x)=4^{x+x}=4^{2x}=(4^2)^x=16^x[/tex]Answer: C,D and E
Related Questions
Hi there,i am having some trouble solving the following two questions relating to extrema and intervals:
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We have the following function:
[tex]y=ln|x^2+x-20|[/tex]The graph of this function is given by:
We know that a function is increasing is the first derivative is greater than zero. The derivative of the given function is given by
[tex]\frac{dy}{dx}=\frac{2x+1}{x^2+x-20}[/tex]Then, the condition is given by
[tex]\frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}>0[/tex]which implies that
[tex]\begin{gathered} 2x+1>0 \\ then \\ x<-\frac{1}{2} \end{gathered}[/tex]By means of this result and the graph from above, f(x) is increasing for x in:
[tex](-5,-\frac{1}{2})\cup(4,\infty)[/tex]Now, the function is decreasing when
[tex]\frac{dy}{dx}<0[/tex]which give us
[tex]\begin{gathered} \frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}<0 \\ 2x+1<0 \\ then \\ x>-\frac{1}{2} \end{gathered}[/tex]Then, by means of this result and the graph from above, the function is decreasing on the interval:
[tex](-\infty,-5)\cup(-\frac{1}{2},4)[/tex]In order to find the local extremal values, we need to find the second derivative of the given function, that is,
[tex]\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{2x+1}{x^2+x-20})[/tex]which gives
[tex]\frac{d^2y}{dx^2}=\frac{(x^2+x-20)(2)-(2x+1)(2x+1)}{(x^2+x-20)^2}[/tex]or equivalently
[tex]\frac{d^2y}{dx^2}=\frac{2(x^2+x-20)-(4x^2+4x+1)}{(x^2+x-20)^2}[/tex]which can be written as
[tex]\frac{d^2y}{dx^2}=\frac{2x^2+2x-40-4x^2-4x-1}{(x^2+x-20)^2}[/tex]then, we get
[tex]\frac{d^2y}{dx^2}=\frac{-2x^2-2x-41}{(x^2+x-20)^2}[/tex]From the above computations, the critical value point is obtained from the condition
[tex]\frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}=0[/tex]which gives
[tex]\begin{gathered} 2x+1=0 \\ then \\ x=-\frac{1}{2} \end{gathered}[/tex]This means that the critical point (maximum or minimum) is located at
[tex]x=-\frac{1}{2}[/tex]In order to check if this value corresponds to a maximum or mininum, we need to substitute it into the second derivative result, that is,
[tex]\frac{d^{2}y}{dx^{2}}=\frac{-2(-\frac{1}{2})^2-2(-\frac{1}{2})-41}{((-\frac{1}{2})^2+(-\frac{1}{2})-20)^2}[/tex]The denimator will be positive because we have it is raised to the power 2, so we need to check the numerator:
[tex]-2(-\frac{1}{2})^2-2(-\frac{1}{2})-41=-\frac{2}{4}+1-41=-40.5[/tex]which is negative. This means that the second derivative evalueated at the critical point is negative:
[tex]\frac{d^2y}{dx^2}<0[/tex]which tell us that the critical value of x= -1/2 corresponds to a maximum.
Since there is only one critical point, we get:
f(x) has a local minimum at x= DNE
f(x) has a local maximum at x= -1/2
The table below summarizes the birth weights to the nearest pound of a sample of 36 newborn babies find the mean birth weight of these 36 babies round your answer to the nearest tenth
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To determine the mean, divide the sum of the products of the number of babies and the birth weight of each by the total number of sample.
Thus, we have the following:
[tex]\bar{x}=\frac{6(5)+7(6)+16(7)+7(8)}{36}[/tex]Simplifying the numerator, we have the following.
[tex]\begin{gathered} \bar{x}=\frac{30+42+112+56}{36} \\ \bar{x}=\frac{240}{36} \end{gathered}[/tex]Therefore, the mean is as follows.
[tex]\bar{x}=6.\bar{6}\approx6.7[/tex]list all congruent pairs of congruent angles and write the ratios of the corresponding side lengths
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Scale factor = 8/6 = 4/3 = 1.33333
Pairs of congruent angles;
m
m
m
Ratio of the coresponding side lengths
AC: BC = LN : MN
4.5 : 6 = 6 : 8
Match the following items.1.(-4, -2)D2.(2, -4)H3.(4, -2)G4.(2, 5)B5.(-1, 1)A6.(-5, 0)F7.(4, 2)E8.(0, -5)CNEXT
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From the graph, we can conclude:
[tex]A=(-4,-2)[/tex][tex]B=(2,-4)[/tex][tex]C=(4,-2)[/tex][tex]D=(2,5)[/tex][tex]E=(-1,1)[/tex][tex]F=(-5,0)[/tex][tex]G=(4,2)[/tex][tex]H=(0,-5)[/tex]Therefore:
Answer:
(-4,-2)----------------->A
(2,-4)------------------>B
(4,-2)------------------>C
(2,5)------------------>D
(-1,1)------------------>E
(-5,0)------------------>F
(4,2)------------------>G
(0,-5)------------------>H
in CDE, J is the centroid. If JF=15 find EJ
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From the figure, J is the centroid. Hence, the lines DH, FE and CG are medians.
Therefore, we can apply the 2/3 rule, that is, the centroid is 2/3 of the way from the vertex to the opposite midpoint.
In other words, we can write
[tex]JE=\frac{2}{3}FE[/tex]since, we know that FE=FJ+JE, we have
[tex]JE=\frac{2}{3}(FJ+JE)[/tex]and, from this equation we can find JE since FJ=15:
[tex]JE=\frac{2}{3}(15+JE)[/tex]The, we obtain
[tex]\begin{gathered} JE=\frac{2}{3}(15)+\frac{2}{3}JE \\ JE-\frac{2}{3}JE=\frac{2}{3}(3\cdot5) \\ \end{gathered}[/tex]in which we moved (2/3)JE to the left hand side and we wrote 15 as 3*5. Now, it reads
[tex]\begin{gathered} \frac{3}{3}JE-\frac{2}{3}JE=2\cdot5 \\ \frac{1}{3}JE=10 \\ JE=3\cdot10 \\ JE=30 \end{gathered}[/tex]Therefore, JE=EJ=30.
(See image below) The point A(-15) is rotated 270° clockwise about the origin. The coordinates of A' are?
Answers
Given:
A (-1,5) rotated 270 clockwise.
Find: A' coordinates
Sol:.
(x,y) Rotated by 270 become (y,-x).
A = (-1,5)
[tex]\begin{gathered} A=(-1,5) \\ A^{\prime}=(5,-(-1)) \\ A\text{'=(5,1)} \end{gathered}[/tex]Bonnie deposits $250 into a new savings account. The account earns 3.5% simple interest per year No money is added or removed from the savings account for 6 years. What is the total amount of money in her savings account at the end of the 6 years?
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Let's begin by identifying key information given to us:
Principal (p) = $250
Interest rate (r) = 3.5% = 3.5/100 = 0.035
Time (t) = 6 years
The simple interest is given by:
[tex]\begin{gathered} A=p(1+rt) \\ A=250(1+0.035\cdot6) \\ A=259(1+0.21) \\ A=250(1.21) \\ A=\text{\$}302.50 \end{gathered}[/tex]Graph the line that has a slope of 1/7 and includes the point (0,5)
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Given the slope 1/7 and point (0,5) we are asked to graph a line.
To do this, the first thing we need to do is to plot the given data.
Plot the given point (0,5)
Next, since we know that the slope of a line is rise/run, given the slope 1/7, it means that from the point (0,5) we will "rise" 1 unit or move 1 unit up, and then "run" 7 units, or move 7 units to the right.
And then, we just connect the two points to form a line
Before sketching the graph, determine where the function has its minimum or maximum value so you can place your first point there.
Answers
We will have the following:
*The zeros in the function are at the values:
[tex]\begin{gathered} -0.5|x-2|=-2\Rightarrow|x-2|=4 \\ x=-2 \\ x=6 \end{gathered}[/tex]So, the zeros are at x = -2 an x = 6.
*The x-intercepts are at the points:
[tex](-2,0)[/tex]And
[tex](6,0)[/tex]*The y-intercept is at the point:
[tex]y=-0.5|0-2|+2\Rightarrow y=1[/tex]So, the y-intercept is located at the point:
[tex](0,1)[/tex]Krista wants to paint her house. she buys 7 1/2. gallons of paint. she uses 3/5 of the paint on the front of her house and then buys 1. 1/2. more gallons of paints how many gallons of paint does she have left
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We know that 7 1/2 is equivalent to 15/2.
If she uses 3/5, then it would remain 9/2 gallons.
[tex]\frac{15}{2}\cdot\frac{3}{5}=\frac{9}{2}=4.5[/tex]Then, she buys 1 1/2, which is equivalent to 1.5 or 3/2.
So, she has 6 gallons of paint.[tex]4.5+1.5=6[/tex]( - 8 - 3) + ( - 12 - 3)withFinding the Slope from Points
Answers
P1 = (-8, -3)
P2 = (-12, -3)
slope = m = (y2 - y1) / (x2 - x1)
Substitution
m = (-3 + 3 ) / (-12 + 8)
Simplification
m = 0 / -4
Result
m = 0
P1 = (7, -9)
P2 = (15, -29)
slope = m = (-29 + 9) / (15 - 7)
m = -20/8
m = -10/4
m = -5/2
This is the graph
ok
2x + 2y = 4
2y = -2x + 4
y = -2/2x + 4/2
y = -x + 2
Sarah can edge a large lawn in 3 hours. Jesse can edge a similar lawn in 2.5 hours. How long would it take Sarah and Jesse if they worked together?
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It takes sarah 3 hours to edge the lawn
so her rate = 1 lawn/ 3 hours
Jesse takes 2.5 hours, so his rate = 1 lawn / 2.5 hours
Together they will take a combined rate of sarah + Jesse
= 1/3 lawn/hour + 1/2.5 lawn/hour
= 1/3 + 1/5/2
= 1/3 + (1 x 2/5)
= 1/3 + 2/5 = 11/15 lawn/hour = 1/r
r = 1/11/15 = 15/11
The time = 15/11 =1 4/11 hours
[tex]\text{The time = 1}\frac{4}{11}hours[/tex]
Answer:
The time = 15/11 =1 4/11 hours
Step-by-step explanation:
Step 1 of 2: Reduce the rational expression to lowest terms x/x^2 - 4xStep 2 of 2: Find the restricted values of X, if any, for the given rational expression.
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We have the following expression:
[tex]\frac{x}{x^2-4x}[/tex]Step 1. Reduce the rational expression to the lowest tem
By factoring the variable x, we get
[tex]\frac{x}{x(x-4)}[/tex]We can cancel x out as long as x is different from zero. Then one restricted value is x=0. So, If x is different from zero, our expression can be reduced to
[tex]\frac{x}{x^2-4x}=\frac{1}{x-4}[/tex]but x must be different from 4.
Step 2. Find the restricted values of x.
Since x can not be zero or four, the restricted values are x=0 and x=4
Help! This problem is way too difficult if you can solve it your a lightsaber!
Answers
Solution
For what value of x is the polynomial above x-axis:
The roots of the graph passes through -3 and 3
so the value of x above will be:
[tex](-3,3)\text{ }U\text{ \lparen3,}\infty)[/tex]The reason why the polynomial is having imaginary roots is
Fundamental Theorem of Algebra we explain that a polynomial will have exactly as many roots as its degree (the degree is the highest exponent of the polynomial). So we know one more thing: the degree is 5 so there are 5 roots in total.
The Fundamental Theorem of Algebra assures us that any polynomial with real number coefficients can be factored completely over the field of complex numbers .
In an experiment involving a treatment applied to 5 test subjects, researchers plan to use a simple random sample of 5 subjects selected from a pool of 7 available subjects. How many different random samples are possible?
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Solution:
The number of ways to select 5 from 7 (when order does not matter) is;
[tex]\begin{gathered} ^7C_5=\frac{7!}{(7-5)!5!} \\ \\ ^7C_5=\frac{7\times6\times5!}{2!\times5!} \\ \\ ^7C_5=21 \end{gathered}[/tex]ANSWER: 21 ways
Adding and subtracting mixed fractions-2/9 - (-3)
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I'm trying to graph the equation:y=1/4x + 3y=2x +10please help
Answers
To graph the equation, the easiest way to get its coordinates is when x = 0 and y = 0.
Let's apply these conditions to get the coordinates of the equation.
a.) y = 1/4x + 3
when,
x = 0 y = 0
y = 1/4(0) + 3 (0) = 1/4x + 3
y = 3 0 - 3 = 1/4x + 3 - 3
-3 = 1/4x
-3(4) = x
-12 = x or x = -12
Thus, the coordinates for equation y = 1/4x + 3 are (0,3) and (-12,0).
Let's now plot the graph of the equation,
The same steps will also be applied to make a graph of equation y = 2x + 10.
There are five performers who represent the Kami access weekend at a comedy club how many different ways are there to schedule this appearances
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SOLUTION
Given the question in the image, the following is the solution to the problem
Step 1: Scheduling n performers is found by n! ways...so five performers gives 5! The different number of ways for 5 performers therefore mean:
[tex]\begin{gathered} 5!=5\times4\times3\times2\times1 \\ =120\text{ ways} \end{gathered}[/tex]Hence, there are 120 different ways of scheduling these appearances.
The variables x = 3/4,y= 2/9 z = 6 are related in such a way that zvaries jointly with x and y.Find z when x = --3 and y = 5.
Answers
Given:
x=3/4, y=2/9 and z=6
z varies jointly with x and y.
From the above data, we can obtain a relation connecting x and z.
The relation connecting z and x can be written as,
[tex]z=8x[/tex]Putting x=3/4 in the above equation,
[tex]z=8\times\frac{3}{4}=6[/tex]So, the relation z=8x is satisfied.
Similarly, the relation between z and y can be written as
Find the length of the missing side on the triangle shown to the right using the Pythagorean theorem.
Answers
The length of the missing side is 68
EXPLANATION
Given:
opposite= 60
Adjacent = 32
Let x be the missing length (hypotenuse)
Using the Pythogaras theorem,
opposite² + adjacent² = hypotenuse²
Substitute the values and evaluate.
60² + 32² = x²
3600 + 1024 = x²
4624 = x²
Take the square root of both-side
√4624 = x
68=x
Hence, the missing side is of length 68.
Point A is located at (-3,5). Find its new coordinates after it is reflected along the x-axis then dilated using a scale factor of 4 with center of dilation at the origin.
Answers
We have (-3,5)
The rule for reflection around the x-axis is
[tex](x,y)\rightarrow(x,-y)[/tex]so the point after the reflection is around x-axis
[tex](-3,5)=(-3,-5)[/tex]for the dilatation we need to multiply the point find above by 4
[tex]A^{\prime}=(-3(4),-5(4))=(-12,-20)[/tex]I need help with this practice problem It has an additional pic of a graph that I will include.
Answers
Given:
The function is given as,
[tex]f(x)=\sin (\frac{\pi x}{2})\text{ . . . . . . (1)}[/tex]The objective is to plot the graph of the function.
Explanation:
To find the maximum point, consider x = 1 in the equation (1),
[tex]\begin{gathered} f(1)=\sin (\frac{\pi(1)}{2}) \\ f(1)=\sin (\frac{\pi}{2}) \\ f(1)=1 \end{gathered}[/tex]Thus, the coordinate is (1,1).
To find the minimum point, consider x = -1 in equation (1).
[tex]\begin{gathered} f(-1)=\sin (\frac{\pi(-1)}{2}) \\ f(-1)=\sin (\frac{-\pi}{2}) \\ f(-1)=-\sin (\frac{\pi}{2}) \\ f(-1)=-1 \end{gathered}[/tex]Thus, the coordinate is (-1,-1).
To plot the graph:
The graph of the function will be,
Hence, the graph of the function is obtained.
write the division expression in words and as a fraction: h ÷ 16
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we have
[tex]h\div16[/tex]in words will be
h between 16
in fraction will be
[tex]\frac{h}{16}[/tex]Solving a distance rate time problem using a system of linear equations
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ANSWER
[tex]\begin{gathered} \text{ Rate of the boat in still water }=36mi\/hr \\ \text{ Rate of the current }=9mi\/hr \end{gathered}[/tex]EXPLANATION
We want to find the rate of the boat in still water and the rate of the current.
Let the boat's rate be x.
Let the rate of the current be y.
When the boat is traveling upstream, it means that it is traveling against the current. This implies that its rate traveling upstream is:
[tex]x-y[/tex]Using the relationship between speed (rate) and distance, we can write that for the upstream travel:
[tex]\begin{gathered} s=\frac{d}{t} \\ \Rightarrow x-y=\frac{108}{3} \\ x-y=36 \end{gathered}[/tex]When the boat is traveling downstream, it means that it is traveling along with the current. This implies that its rate traveling downstream is:
[tex]x+y[/tex]and:
[tex]\begin{gathered} x+y=\frac{108}{2} \\ x+y=54 \end{gathered}[/tex]Now, we have two simultaneous linear equations:
[tex]\begin{gathered} x-y=36 \\ x+y=54 \end{gathered}[/tex]Solve for x by elimination. To do this, add the two equations and simplify:
[tex]\begin{gathered} x-y+x+y=36+54 \\ 2x=90 \\ x=\frac{90}{2} \\ x=45mi\/hr \end{gathered}[/tex]Solve for y by substituting x into the second equation:
[tex]\begin{gathered} 45+y=54 \\ \Rightarrow y=54-45 \\ y=9mi\/hr \end{gathered}[/tex]Hence, the rate of the boat in still water is 36 mi/hr and the rate of the current is 9 mi/hr.
Select One 3^6/3^3 ↑ Select One Add Subtract Multiply can you help me with this please?[tex] {3}^{6} {3}^{3} [/tex]thats supposed to be a fraction
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Find 3^6 / 3^3
Operation is SUBSTRACT
Let 3 equal base
and SUBSTRACT (6-3) = 3
THEN result is
3^(6-3) = 3^3
ANSWER IS SUBSTRACT
then Multiply 3x3x3 = 27
I need help with this practice I attempted this practice previously and my attempt is in the picture
Answers
We will draw a sketch for the given triangle to find its area
The area of the triangle will be
[tex]A=\frac{1}{2}\times XY\times ZM[/tex]Since ZX = ZY = 7, then the triangle is isosceles
Then the height ZM will bisect the base XY
Then we can find ZM by using Pythagoras Theorem
[tex]\begin{gathered} ZM=\sqrt[]{7^2-3^2} \\ ZM=\sqrt[]{49-9} \\ ZM=\sqrt[]{40} \\ ZM=2\sqrt[]{10} \end{gathered}[/tex]Since XY = 6, then
The area of the triangle is
[tex]\begin{gathered} A=\frac{1}{2}\times6\times2\sqrt[]{10} \\ A=6\sqrt[]{10}\text{ square units} \end{gathered}[/tex]How do I rearrange 2x + y = -3 Into a Y = mx + b
Answers
Solution
Given
[tex]2x+y=-3[/tex]Move the term 2x to the right-hand side and change its sign
[tex]y=-2x-3[/tex]Therefore, the required answer is
[tex]y=-2x-3[/tex]Each participant must pay $14 to enter the race. Each runner will be given a T-shirt that cost race organizers $3.50. If the T-shirt was the only expense for the race organizers, which of the following expressions represents the proportion of the entry fee paid by each runner that would be donated to charity? is it $14.00÷($14.00-$3.50)
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From the statement of the problem, we know that each participant:
• pays $14 to enter the race,
,• receives a T-shirt that cost $3.50 to the organizers.
The earnings minus the cost of the T-shirts will be donated to charity, so for each participant, the donation will be $14 - $3.50. So the proportion of the entry fee paid by each runner that would be donated to charity is:
[tex]\frac{14.00-3.50}{14.00}[/tex]Answer
A. ($14.00 - $3.50) / $14.00
Determine the amount of the ordinary annuity at the end of the given period. (Round your final answer to two decimal places.)$200 deposited quarterly at 6.9 for 6 years
Answers
For solving this question it is necessary to apply the formula
[tex]FV=P\cdot(\frac{(1+r)^n-1}{r})[/tex]Where:
FV = future value of the account;
P= deposit = $200
r = quarterly percentage - use decimal=0.069
n = number of deposits = 4* 6=24
[tex]\begin{gathered} FV=P\cdot(\frac{(1+r)^n-1}{r}) \\ FV=200\cdot(\frac{(1+\frac{0.069}{4})^{24}-1}{\frac{0.069}{4}}) \\ FV=200\cdot(\frac{(1+0.01725)^{24}-1}{0.01725}) \\ FV=200\cdot(\frac{(1.01725)^{24}-1}{0.01725}) \\ FV=200\cdot\frac{0.5075}{0.01725}=5884.38 \\ FV=5884.38 \end{gathered}[/tex]FV=$5884.38
I need help with these 3 questions on this math problem
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The table represents Monte's savings
The rate of change is 8. This means that he earns 8 dollars per hour
The graph represents Ramon's savings
The rate of change is 6. This means that he earns 6 dollars per hour
The answers to the questions are shown below
1) The rates of change represent the amount that each person earns per hour.
2) The rate of change of Monte's is greater. It means that he earns more per hour than Ramon.
3) We would find the equation in slope intercept form representing each person's savings. The equation of a line in the slope intercept form is expressed as
y = mx + c
where
m = slope
c = y intercept
The y intercept is the value of y when x = 0. On the graph, it is the value of y at the point where the line cuts the vertical axis. We have
For the table,
when x = 0, y = 3
Thus, c = 3
m = 8
The equation for Monte's total savings is
y = 8x + 3
After 8 working hours, x = 8. We would substitute x = 8 into the equation. We have
y = 8 * 8 + 3 = 64 + 3
y = 67
Monte's total savings after 8 hours is $67
For the graph, c = 6
m = 6
The equation for Ramon's total savings is
y = 6x + 6
After 8 working hours, x = 8. We would substitute x = 8 into the equation. We have
y = 6 * 8 + 6 = 48 + 6
y = 54
Ramon's total savings after 8 hours is $54