Simplify the result if possible assume all variables represent positive real numbers
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The function [tex]log_{b} \sqrt[3]{(\frac{x^{8} }{y^{9} z^{6} })}[/tex] is simplified to be [tex]8/3log_{b}x-3log_{b}y-2log_{b}z[/tex]
How to simplify the functionThe function is simplified using the laws of logarithm
[tex]log_{b} \sqrt[3]{(\frac{x^{8} }{y^{9} z^{6} })}[/tex]
[tex]= log_{b}(\frac{x^{8} }{y^{9} z^{6} })^{1/3}[/tex]
[tex]= log_{b}(\frac{x^{8/3} }{y^{9/3} z^{6/3} })[/tex]
[tex]= log_{b}(\frac{x^{8/3} }{y^{3} z^{2} })[/tex]
Applying the quotient rule
[tex]= log_{b}x^{8/3}-log_{b}( y^{3} z^{2})[/tex]
Applying the product rule
[tex]= log_{b}x^{8/3}-(log_{b}y^{3}+log_{b}z^{2})[/tex]
expanding the parenthesis
[tex]= log_{b}x^{8/3}-log_{b}y^{3}-log_{b}z^{2}[/tex]
Applying the exponential rule
[tex]= 8/3log_{b}x-3log_{b}y-2log_{b}z[/tex]
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Related Questions
⦁ The total cost after tax to repair Deborah’s computer is represented by 0.08(50h)+50h, where h represents the number of hours it takes to repair Deborah’s computer. What part of the expression represents the amount of tax Deborah has to pay? Explain.
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The part of the expression that represents the amount of tax Deborah will pay is 0.08(50h).
What part of the expression represents taxes?The given expression is known as a linear equation. A linear expression is an expression that has a single variable raised to the power of 1.
Tax is the compulsory amount levied by on goods and services. Tax is usually a percentage of the price of a good and service. Taxes increases the price of a good or service. The tax paid would increase with the total income earned.
Tax = tax percentage x total cost
Tax = 0.08 x (total hours worked x cost per hour)
Tax = 0.08 x (50 x h)
Tax = 0.08(50h)
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Identify the vertex, intercepts and whether of the graph of the function below opens up or down. Type your answers as a point (x,y). If an intercept does not exist type "none". If more than one intercept exists you can type either intercept.f(x)= -|x-9|+16 Vertex = Answerx intercept = Answery intercept = Answergraph opens Answer
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The absolute value function :
[tex]f(x)=\pm\lvert x-h\rvert+k[/tex]has a vertex at (h, k) and it opens upward if the sign before the absolute value sign is positive. It open downward if the sign is negative.
From the problem, we have :
[tex]f(x)=-\lvert x-9\rvert+16[/tex]The vertex will be (9, 16)
x intercept is the value of x when f(x) = 0.
Set f(x) = 0 and solve the value of x.
[tex]\begin{gathered} 0=-\lvert x-9\rvert+16 \\ \lvert x-9\rvert=16 \end{gathered}[/tex]In solving absolute values, you will get two values, one for the positive and one for the negative.
[tex]\begin{gathered} x-9=16 \\ x=16+9 \\ x=25 \end{gathered}[/tex][tex]\begin{gathered} x-9=-16 \\ x=-16+9 \\ x=-7 \end{gathered}[/tex]The x-intercepts are (-7, 0) and (25, 0)
y-intercept is the value of f(x) when x = 0.
Set x = 0, and evaluate f(x)
[tex]\begin{gathered} f(x)=-\lvert0-9\rvert+16 \\ f(x)=-9+16 \\ f(x)=7 \end{gathered}[/tex]The y-intercept is (0, 7)
The sign before the absolute value sign is negative, so it opens downward
Tracy wants to buy some food for her slumber party. Great Foods Grocery Store is selling 2 bags of potato chips for $6.50 and 5 two-liter sodas for $3.00. Best FoodsGrocery Store is selling 2 bags of potato chips for $5.00 and 3 two-liter sodas for $2.00. if p = potato chips and s= two-liter sodas, write a system of equations to model thisproblem2p=5.00 and 3s=2.006.50p + 3.00s= 7 and 5p +2s=52p=6.50 and 5s=3.002p + 5s 9.50 and 2p +39 7.00
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Answer:
[tex]\begin{gathered} \text{ The system that describes this problem:} \\ 2p+5s=9.50 \\ 2p+3s=7.00 \end{gathered}[/tex]Step-by-step explanation:
Let p be the potato chips
Let s be the two-liter sodas
Then, if Great Foods Grocery Store is selling 2 bags of potato chips for $6.50 and 5 two-liter sodas for $3.00:
[tex]2p+5s=9.50[/tex]For Best Foods Grocery Store:
2 bags of potato chips for $5.00 and 3 two-liter sodas for $2.00.
[tex]2p+3s=7[/tex]I need to know the scale factor and what S is.
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In order to find the scale factor between the triangles, we can compare the sides PR and PT, which are corresponding sides between the triangles.
The side PR has a length of 27 units, and the side PT has a length of 9 units, so we can find the scale factor by dividing one length by the other:
[tex]\text{scale factor}=\frac{PR}{PT}=\frac{27}{9}=3[/tex]Now that we have the scale factor, we can find the length of PS by comparing it with the corresponding side PQ:
[tex]\begin{gathered} \text{scale factor}=\frac{PQ}{PS} \\ 3=\frac{24}{PS} \\ PS=\frac{24}{3}=8 \end{gathered}[/tex]If the length of PS is 8 units and S is above the x-axis to the right, its coordinates will be (8, 0).
Not a timed or graded assignment. Need full work shown. Quick answer with work = amazing review
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The Solution:
Given the expression below:
[tex]10\text{ }\sqrt[]{112m^6}[/tex]We are asked to simplified in radical form.
Let's find the factors of 112.
[tex]\begin{gathered} 112=2\times56 \\ =2\times2\times28 \\ =2\times2\times2\times14 \\ =2\times2\times2\times2\times7 \end{gathered}[/tex][tex]\sqrt[]{m^6}=\sqrt[]{m^3\times m^3}[/tex]So,
[tex]\sqrt[]{112m^6}=\sqrt[]{112\times m^3\times m^3}=\sqrt[]{2\times2\times2\times2\times7\times m^3\times m^3}[/tex][tex]10\sqrt[]{112m^6}=10\sqrt[]{2\times2\times2\times2\times7\times m^3\times m^3}=10(2\times2\times m^3)\text{ }\sqrt[]{7}[/tex]Thus,
[tex]10\sqrt[]{112m^6}=\text{ }10(2\times2\times m^3)\text{ }\sqrt[]{7}=10(4)m^3\text{ }\sqrt[]{7}=40m^3\text{ }\sqrt[]{7}[/tex]Therefore, the correct answer is
[tex]40m^3\text{ }\sqrt[]{7}[/tex][tex](1-(-1)*2)x^{2} -1[/tex]
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Answer:
[tex]3x {}^{2} - 1[/tex]
Step-by-step explanation:
(1-(-1)×2) x² - 1
When there is a - in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses.
(1+1x2)x2-1
Any expression multiplied by 1 remains the same
(1+2) x²-1
Add the numbers
3x2-1
Which of the following Polynomial does not have real roots?
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Using the formula below
[tex]\begin{gathered} b^2-4ac \\ \text{Solving for the first option} \end{gathered}[/tex]For 2x²+4x+4, relataing it to the general form of a quadratic equation,
[tex]ax^2+bx+2=0[/tex]Where a, b, and c for the 2x²+4x+4 is given below,
[tex]\begin{gathered} a=2,b=4,c=4 \\ b^2-4ac=4^2-4(2\times4)=16-32=-16 \end{gathered}[/tex]Since b²-4ac is not supposed to be negative (i.e complex roots),
Answer is A
2 hours reading1 hour playing with friendsRate ? Ratio only ? Or a proportion?
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Answer:
[tex]\text{Ratio Only}[/tex]Explanation:
We get a rate when we consider how a quantity changes in relation to another
When we compare two quantities, we have a ratio
Equating two ratios, we have a proportion
From the question, we are comparing the number of reading hours to the number of playing hours
That means we have a ratio only
Determine whether each triangle is isosceles, equilateral, or can not be determined.
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We have the following:
An isoceles triangle is that it has two equal sides and an equilateral triangle is that all sides are equal.
A.
It has two equal sides, therefore it is an isoceles triangle
B.
It has two equal sides, therefore it is an isoceles triangle
C.
It is not possible to determine since an angle of 60 ° does not determine the size of the other two angles
D.
It has two equal angles, which means that it has two equal sides, therefore it is an isoceles triangle.
Question 7(Multiple Choice Worth 2 points)
(Angle Relationships MC)
Two angles are complementary to each other. One angle measures 32°, and the other angles measure (12x-20) determine the value of x
O 64
O 6.5
32.5
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If the sum of (12x - 20)° + 32° = 180° then value of x is 14°.
What are supplementary and complementary angles?When the sum of the two angles is 180° they are said to be supplementary.
When the sum of the two angles is 90° they are said to be complementary.
Given, Two angles are complementary to each other.
One angle measures 32°, and the other angles measure (12x - 20)°.
∴ (12x - 20)° + 32° = 180°.
12x - 20° + 32° = 180°.
12x = 180° - 12°.
12x = 168°.
x = 14°.
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mmhey again um yea i looked at my notes but thats not really helpful :/
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We have the following:
Since the sides are equal, it means that all the angles are equal, we also know that the sum of the 3 angles within a triangle is always 180°, therefore
[tex]\begin{gathered} 5x+5x+5x=180 \\ \text{solving for x:} \\ 15x=180 \\ x=\frac{180}{15} \\ x=12 \end{gathered}[/tex]The value of x is 12
changing the value of b in f(x)=mx+b results in a translation or reflection?
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2. Which statement is an example of a transitive relationship? If ctm and m || n, then cin. If x = 2y and 2y 8, then x = 4. If a Il band b || c, then a || o. If min and mlp, then m || p.
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A relationship is said to be transitive, if
a R b, b R c, then → a R c.
Test the given options
For the first option, if x = 2y and 2y = 8, then for transitive relationship,
[tex]\begin{gathered} x=2y \\ 2y=8 \\ then,x=8 \end{gathered}[/tex]the first option is not correct because x ≠ 4
For the second option, If a Il b and b || c, then a || c
[tex]\begin{gathered} a\text{ R b means that a is parallel to b} \\ b\text{ R c means that b is parallel to c} \\ a\text{ R c means that a is parallel to c} \end{gathered}[/tex]Looking at the second option, there is a relationship of parallelism between a, b and c, therefore, this is a transitive relationship
For the third option
If m ⊥ n and m ⊥ p, then m ∥ p.
The statement from the point of view of transitive relationship is incorrect
it should be, n ⊥ p.
The data points on the scatter plot below show the amount of natural snow received and the amount of artificial snow made by a ski resort for each of 25 weeks. Draw the line of best fit for these data points. (It doesn't have to be the exact line of best fit. Just draw your best approximation.)
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We need to draw the line of best fit for the data points in the scatter plot.
One way of drawing this line is by leaving approximately the same amount of points above and below it:
Notice that the line has to follow the trend of the points: the amount of artificial snow made decreases as the amount of natural snow received increases.
Determine the correct order of the numbers from least to greatest. 1.3, -2.875, 6.75, -4, -1.67, -3.75, 3.5
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By definition, the Positive numbers are those numbers greater than zero and Negative numbers are those numbers less than zero.
Therefore, you know that the Positive numbers are greater than the Negative numbers.
For this exercise it is also important to remember that the Absolute value of a number tells you its distance from zero on the Number line. The Absolute value of a number is always positive.
Knowing the above, you can set up that:
[tex]-4<-3.75<-2.875<-1.67<1.3<3.5<6.75[/tex]Remember that this symbol means "Less than":
[tex]<[/tex]The answer is:
[tex]-4,-3.75,-2.875,-1.67,1.3,3.5,6.75[/tex]Find the surface area of the cylinder below. Use 3.14 for l~l. round your answer to the nearest tenth.
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Which I two triangles are congruent? Complete the congruence statement.
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Answer:
∆vuw≈∆bca∆uvw≈∆cba∆vwu≈∆bacPut the following equation of a line into slope-intercept form, simplifying all fractions. Žy 2x = 8 Answer: Submit Answer attempt 1 out of 2 Privacy Policy Terms of Service
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We have the following:
the equation of a line into slope-intercep form is:
[tex]y=mx+b[/tex]now,
[tex]\begin{gathered} 2y-2x=8 \\ 2y=2x+8 \\ y=x+4 \end{gathered}[/tex]therefore, the answer is:
[tex]y=x+4[/tex]given f(x) = 5x-1 , evaluate f(5)= also solve f(x)=14 then x= ?
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You have the following expression for f(x):
f(x) = 5x - 1
In order to find the value of f(5), replace x = 5 into the previous expression and simplify:
f(5) = 5(5) - 1 = 25 - 1 = 24
Hence, f(5) is 24
To determine the value of x when f(x) = 14, solve for x, as follow:
14 = 5x - 1 add 1 both sides
14 + 1 = 5x
15 = 5x divide by 5 both sides
15/5 = x
3 = x
x = 3
Hence, x = 3 when f(x) = 14
Which solution value satisfies the inequality x – 6 ≥ –7?Question 4 options:A)x = –7B)x = 0C)x = –5D)x = –2
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Let's solve the inequality:
[tex]\begin{gathered} x-6\ge-7 \\ x\ge-7+6 \\ x\ge-1 \end{gathered}[/tex]Hence any solution of the inequality has to be equal or greater than -1. From the list given we notice that the only value that fulfills this condition is x=0, therefore the correct option is B.
Which equation is nonlinear? x=-4 y= 0 y= 2/3x- 2 y= ײ +1
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equationWhich equation is nonlinear?
x=-4
y= 0
y= 2/3x- 2
y= ײ +1
__________________
Linear eqaution form
y = mx +b
x=-4 (Line)
y= 0 (lineon the axis)
y= 2/3x- 2 (This is a linear equation )
_____________________
Answer
y= ײ +1
Determine the linear equation of the vertical and horizontal line passing through the point (5,8).
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Given:
The horizontal and vertical line psses through the point (5,8).
To find:
Find the equation of vertical and horizontal line passing through the given point.
Equation of vertical line:
[tex]x=5[/tex]Equation of horizontal line:
[tex]y=8[/tex]Which of the following has the same value as cos 2pi/3
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First, let's calculate the value of cos 2pi/3:
[tex]\cos\frac{2\pi}{3}=-0.5[/tex]Now, let's calculate the value of each option:
[tex]\begin{gathered} \cos\frac{\pi}{6}=0.866\\ \\ \\ \\ \cos\frac{4\pi}{3}=-0.5\\ \\ \\ \\ \sin\frac{5\pi}{3}=-0.866\\ \\ \\ \\ \sin\frac{7\pi}{6}=-0.5\\ \\ \\ \\ \cos\frac{11\pi}{6}=0.866 \end{gathered}[/tex]Therefore the correct options are B and D.
The table shows the times Allison clocked in and out of work this week. Her employer rounds to the nearest quarter hour. How many total hours did she work?Time in. Time out8:03 am. 5:13 pm7:58 am. 4:49 pm7:46 am. 4:41 pm8:23 am. 4:50 pm7:31 am. 4:32 pmA) 44.25 hoursB) 45.5 hours C) 42.5 hours D) 44 hours
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Answer:
A) 44.25 hours
Explanation:
To round time to the nearest quarter-hour, round those times within 7 minutes of a 15-minute mark to that 15 minutes.
Thus, the time in and time out are rounded to the nearest quarter-hour in the table below:
Next, add the differences:
[tex]\begin{gathered} Total=9:15+8:45+9:00:8:15+9:00 \\ =43hours+(15+45+15)minutes \\ =44\text{ hours 15 minutes} \\ =44+\frac{15}{60} \\ =44.25\text{ hours} \end{gathered}[/tex]She worked a total of 44.25 hours.
Option A is correct.
Three vertices of parallelogram JKLM are J(1, 4), K(5, 3), and L(6,-3). Find the coordinates ofvertex M.The coordinates are MOD.
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Given data:
The coordinate of first vertex is J(1, 4).
The coordinate of second vertex is K(5, 3).
The coordinate of third vertex is L(6,-3).
Assume the coordinate of M is (x, y).
The diagonal of the parallelogram intersect at the mid point, the mid point of the diagonal JL is,
[tex]\begin{gathered} a=\frac{1+6}{2} \\ =\frac{7}{2} \\ b=\frac{4-3}{2} \\ =\frac{1}{2} \end{gathered}[/tex]This is also the mid point of KM diagonal.
[tex]\begin{gathered} \frac{7}{2}=\frac{x+5}{2} \\ x=2 \\ \frac{1}{2}=\frac{y+3}{2} \\ y=-2 \end{gathered}[/tex]Thus, the coordinate of M is (2, -2).
In a raffle, one ticket will win a $930 prize, and the other tickets will win nothing. There are 500 in the raffle, each costing $6. If you buy a ticket, what is the expected profit?
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Find all values of y such that the distance between (5,y) and (-7,2) is 18.
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Find all values of y such that the distance between (5,y) and (-7,2) is 18.
Remember that
The formula to calculate the distance between two points is equal to
[tex]d=\sqrt[]{(y2-y1)^2+(x2-x1)^2}[/tex]substitute the given values
[tex]18=\sqrt[]{(y-2)^2+(5+7)^2}[/tex][tex]18=\sqrt[]{(y-2)^2+144}[/tex]squared both sides
[tex]18^2=(y-2)^2+144[/tex]solve for y
[tex]\begin{gathered} (y-2)^2=324-144 \\ (y-2)^2=180 \end{gathered}[/tex]take square root on both sides
[tex]\begin{gathered} y-2=\pm\sqrt[]{180} \\ y=2\pm\sqrt[]{180} \end{gathered}[/tex]simplify
[tex]y=2\pm6\sqrt[]{5}[/tex]Describe two methods you could use to solve for `x` in `1.12^{x}=20`
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EXPLANATION:
Given;
We are given the following equation;
[tex]1.12^x=20[/tex]Required;
We are required to describe two methods which can be used to solve for x in this equation.
Step-by-step solution;
We can solve for the variable x by taking the natural log of both sides of the equation. This is shown below;
[tex]1.12^x=20[/tex]We take the natural log of both sides;
[tex]ln1.12^x=ln20[/tex]Next we apply the log rule;
[tex]\begin{gathered} If: \\ log_bx^a \\ Then: \\ alog_bx \end{gathered}[/tex]Therefore, our equation is now refined and becomes;
[tex]xln1.12=ln20[/tex]Divide both sides by ln(1.12);
[tex]x=\frac{ln(20)}{ln(1.12)}[/tex]A second method is to express it as a logarithmic equation;
[tex]1.12^x=20[/tex]We shall apply the log rule which is;
[tex]\begin{gathered} If: \\ log_bx=a \end{gathered}[/tex][tex]\begin{gathered} Then: \\ b^a=x \end{gathered}[/tex]For example;
[tex]\begin{gathered} If: \\ log_{10}100=2 \end{gathered}[/tex][tex]\begin{gathered} Then: \\ 10^2=100 \end{gathered}[/tex]Therefore, for the equation given;
[tex]\begin{gathered} If: \\ 1.12^x=20 \end{gathered}[/tex][tex]\begin{gathered} Then: \\ log_{1.12}20=x \end{gathered}[/tex]Note that both solutions can be simplified eventually with the use of a calculator.
ANSWER:
(1) By taking the natural log of both sides
(2) By expressing the equation as a logarithmic equation
Which answer describes the pattern in this sequence?12) 412, 1, 21OaddO multiply by 2o subtract 1multiply by12
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Answer:
The answer that describes the pattern in this sequence is;
[tex]\text{ multiply by }\frac{1}{2}[/tex]Explanation:
Given the sequence;
[tex]2,1,\frac{1}{2},\frac{1}{4},\ldots[/tex]The sequence shows a Geometric Progression.
The common ratio of the sequence would be;
[tex]\begin{gathered} r=\frac{1}{2}=\frac{\frac{1}{2}}{1}=\frac{\frac{1}{4}}{\frac{1}{2}} \\ r=\frac{1}{2} \end{gathered}[/tex]Therefore, the answer that describes the pattern in this sequence is;
[tex]\text{ multiply by }\frac{1}{2}[/tex]
A park volunteer Plans to work on pallet Stonewall for one hour every Monday one every Wednesday and three hours on Friday
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Solution
For this case we can do the following:
Part a
we can select the points (0,0) and (2,10) and k is given by:
[tex]k=\frac{10-0}{2-0}=5[/tex]Part b
The equation is given by:
y= 8x
Part c
We can replace x= 16 in the equation and we got:
y= 8*16= 128 hours