Write the expression as a single logarithm, and simplify the result, if possible.
Answers
To simplify the logarithmic expression, we need to remember some logarithm rules. Some of the rules are:
[tex]\begin{gathered} \log _b(x\cdot y)=\log _bx+\log _by \\ \log _b\frac{x}{y}=\log _bx-\log _by \end{gathered}[/tex]In the expression that we have, we can group it into (log₃ 54 + log₃ 10) - log₃ 20.
We can apply the logarithm product rule in this expression: log₃ 54 + log₃ 10. This becomes log₃ (54 x 10) = log₃ 540
[tex]\begin{gathered} \log _bx+\log _by=\log _b(x\cdot y) \\ \log _354+\log _310=\log _3(54\cdot10)=\log _3540 \end{gathered}[/tex]Then, we can apply the logarithm quotient rule in this expression log₃ 540 - log₃ 20. This becomes log₃ (540/20) = log₃ 27 which is equal to 3.
[tex]\begin{gathered} \log _bx-\log _by=\log _b\frac{x}{y} \\ \log _3540-\log _320=\log _3\frac{540}{20}=\log _327=3 \end{gathered}[/tex]The single logarithm is:
[tex]\log _3\frac{54\cdot10}{20}[/tex]which then simplified to:
[tex]\log _327[/tex]which is equal to 3.
Related Questions
x2 + 2x - 1 = 0 solve for x
Answers
Answer:
Ohio state university football roster
What is the slope of the line?
Answers
Answer: 2
Step-by-step explanation: For every time it goes right 1 it goes up 2
Answer: 2
Step-by-step explanation:
Find two points, and then do rise/run. (0,-3) and (1,-1) are both points, so rise over run is 2/1, so your slope is 2
The diameter of a circle is 10 yards. Finthe approximate circumference of thecircle, using 3.14 for PI.
Answers
The circumference is defined by the formula below.
[tex]C=\pi d[/tex]Where d is the diameter.
Let's replace the diameter and pi.
[tex]C=3.14\cdot10=31.4yd[/tex]Therefore, the circumference is 31.4 yards.If n=12, ¯x (x-bar)=34, and s=19, construct a confidence interval at a 90% confidence level. Assume the data came from a normally distributed population.Give your answers to one decimal place. < μ <
Answers
n=12
s=19
X=34
ConfLevel=90%
A confidence level of 90%
represent a Z statistical of 1.645
the Confidence interval is given by
[tex]X\pm Z*\frac{s}{\sqrt{n}}[/tex]then
[tex]34\pm1.645*\frac{19}{\sqrt{12}}[/tex][tex]34\pm9.02254[/tex]then the interval is
[tex]25.0\leq u\leq43.0[/tex]On a cold day the temperature is a certain change -34.08 over 4.8 hours what was the average change in temperature per hour 
Answers
The average change in temperature per hour is -7.1 degree/hour.
What is average rate of change? It is the average amount by which the function changed per unit throughout that time period.Divide the change in y-values by the change in x-values to find the average rate of change. The average rate of change is especially useful for determining changes in measurable values such as average speed or average velocity. Here are some examples of average rates of change: A bus travels at an average speed of 80 kilometers per hour. A lake's fish population grows at a rate of 100 per week. When the voltage in an electrical circuit drops by one volt, the current in the circuit drops by 0.2 amps.Given,
Change in temperature = -34.08 degree
Total time = 4.8 hours
Average change = -34.08 / 4.8
= -7.1 degree/hour
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Which of the following is an irrational number?A. VoB. 73C. 1.36D. -0.19
Answers
We have to identify the irrational number.
An irrational number is a number that can not be expressed as a fraction.
The first option is the square root of 0 which is equal to 0. It is an integer so it can be expressed as a fraction.
Then, it is not an irrational number.
The second option is the square root of 3. This does not have a solution that can be expressed as a fraction or a number with a finite number of decimals.
Then, it can be considered an irrational number.
The third option is a periodic decimal. They can be expressed as fractions so they are not irrational numbers.
the fourth option is a negative decimal number, which can be expressed as decimal, like -19/100. Then, it is not an irrational number.
Answer: the only irrational number is √3 [Option B]
Paulina’s income from a job that pays her a fixed amount per hour is shown in the graph. Use the graph to find the total income earned for working four 8-hour days all at the standard rate.
Answers
Solution
We need to find the equation of an income y at any time x
Using two-point formula
[tex]\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}[/tex]At points (1, 10) and (6, 60)
[tex]\begin{gathered} \Rightarrow\frac{y-10}{x-1}=\frac{60-10}{6-1}=\frac{50}{5}=10 \\ \\ \Rightarrow\frac{y-10}{x-1}=10 \\ \\ \Rightarrow y-10=10x-10 \\ \\ \Rightarrow y=10x \end{gathered}[/tex]Four 8 hours = 8 + 8 + 8 + 8 = 32
At x = 32 => y = 10 × 32 = 320
Therefore, the total income for working four 8 hours is $320
11) What is probability of not being born in a month that starts with avowel?
Answers
total number of months: 12
Months that don't start with a vowel: 9
January
February
March
May
June
July
September
November
December
probability of not being born in a month that starts with a
vowel ( or being born in a month that starts with a consonant)
9/12 = 0.75
is 3 a factor of 12
Answers
Answer:YES
Step-by-step explanation:
Answer:
Yes
Step-by-step explanation: the factors of 12 are 1,2,3,4,6 and 12
Systems of equations
Answers
The slopes to the linear functions are given as follows:
2. Parallel line: slope of m = -3.
3. Two points: slope of m = -0.06.
What is the slope of a linear function?A linear function is modeled according to the following rule:
y = mx + b.
The coefficient m represents the slope of the linear function, which is the rate of change, given by change in y divided by change in x.
When two functions are parallel, they have the same slope. In item 2, the function is defined as follows:
-7x - 2y = 6.
In slope-intercept format, the function is given by:
2y = -7x - 6
y = -3.5x - 3.
Hence the slope of the parallel line is of -3.
Given two points, the slope is given by change in y divided by change in x. For problem 3, the two points are given as follows:
(-7,0) and (9,-1).
Hence the slope is given by:
m = (-1 - 0)/(9 - (-7)) = -1/16 = -0.06.
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Find the value of X.
Answers
Answer:
15
Step-by-step explanation:
Find the y-intercept of the following line. y=14/17 x+14
Answers
The y-intercept of the line y = 14 / 17 x + 14 is 14.
How to find the y-intercept of a line?The equation of a line can be represented in different form such as slope intercept form, point slope form, standard form and general form.
Therefore, let's represent it in slope intercept form.
y = mx + b
where
m = slopeb = y-interceptTherefore, using the equation of a line in slope intercept form, the y-intercept of the equation y = 14 / 17 x + 14 is 14.
The y-intercept of a line is the value of y when x = 0.
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Determine the value(s) for which the rational expression 8q+8/3q2−q−14 is undefined.
Answers
The required, for q = -2 and q = 3/7 the given rational expression is not defined.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given expression,
= 8q+8/3q²−q−14
Simplifying
=8(q+1)/3q² -7q + 6q - 14
= 8 (q + 1)/ q(3q - 7) + 2(3q - 7 )
= 8(q + 1) / q(3q - 7)(q + 2)
For q = -2 and q = 3/7 the given rational expression is not defined.
Thus, For q = -2 and q = 3/7 the given rational expression is not defined.
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1. Charlene wants to center a rectangular pool in her backyard so that the edges of the pool are an equal distance from the edges of the yard on all sides. The yard currently measures 60 m by 50 m. She wants to use ½ of the area of the yard for the pool. Create an equation for the pool’s dimensions and solve for the distance the pool is from the edge of the yard. Round your final answer to the nearest tenth of a meter.
Answers
First, we have to calculate the length of the sides of the pool, we are told that the scale factor of the backyard to the pool size equals 1/2, then we can find the length of the sides of the pool by multiplying the lengths of the sides of the backyard by 1/2, like this:
length of the pool = length of the yard * 1/2
width of the pool = width of the yard * 1/2
By replacing the 60 m for the length of the yard and 50 m for the width, we get:
length of the pool = 60 * 1/2 = 30
width of the pool = 50 * 1/2 = 25
Let's call x1 to the distance from the base of the pool to the bottom side of the yard and x2 to the distance from the top side of the pool to the top side of the yard, then we can formulate the following equation:
width of the yard = width of the pool + x1 + x2
Since we want the edges to be at an equal distance, x1 and x2 are the same, then we can rewrite them as x:
width of the yard = width of the pool + x + x
width of the yard = width of the pool + 2x
Replacing the known values:
60 = 30 + 2x
From this equation, we can solve for x to get:
60 - 30 = 30 - 30 + 2x
30 = 2x
30/2 = 2x/2
15 = x
x = 15
Now, let's call y1 to the distance from the right side of the corresponding side of the yard and y2 to the distance from the left side of the pool to the left side of the yard, with this, we can formulate the following equation:
length of the yard = length of the pool + y1 + y2
Since we want the edges to be at an equal distance, y1 and y2 are the same, then we can rewrite them as y:
length of the yard = length of the pool + y + y
length of the yard = length of the pool + 2y
Replacing the known values:
50 = 25 + 2y
50 - 25 = 25 - 25 + 2y
25 = 2y
25/2 = 2y/2
12.5 = y
y = 12.5
Now, we know that the pool must be at a distance of 15 m from the horizontal sides of the pool to the horizontal sides of the yard and that it must be at a distance of 12.5 m from the vertical sides of the pool to the vertical sides of the yards.
Here is a figure that depicts the results:
Write this ratio as a fraction in simplest form without any units.
Answers
The ratio as a fraction in simplest form
without any units 56 days to 5 week is 8:5
what is Ratio ?
A ratio is non - zero ordered pair of numbers a and b written as a a/b.
A proportion is a mathematical expression in which two ratio are specified to be equal .
A fraction is represented by p/q where q≠ 0
it is asked in the question to write the ratio as a fraction in simplest form without any units 56 days to 5 week .
the fraction its simplest form means to write in the ratio where it cannot be further divided from each other .
1 week = 7 days
5 week = 35 days
the ration of 56 days : 35 week
56:35
8:5
therefore the ratio as a fraction in simplest form without any units 56 days to 5 week is 5:8.
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f(x)= 5x-3
g(x)= 2x+4 /2
solve for f(2)
1. f^1(2)
2. f^1 (g) (2)
3. f (g^1) (2)
Answers
The solutions are;
1. f⁻¹ (2) = 1
2. f⁻¹ (g(2)) = 7 / 5
3. f (g⁻¹(2)) = - 3
What is mean by Function?
A relation between a set of inputs having one output each is called a function.
Given that;
The function are,
⇒ f (x) = 5x - 3
And, g (x) = (2x + 4) / 2
Now,
The solution of the functions are;
Since, The function is;
f (x) = 5x - 3
To find the inverse of the above as;
f (x) = 5x - 3
y = 5x - 3
Solve for x as;
y + 3 = 5x
x = (y + 3) / 5
Substitute x = f⁻¹ (x) we get;
f⁻¹ (x) = (x + 3) / 5
1. So, Substitute x = 2 as;
f⁻¹ (x) = (x + 3) / 5
f⁻¹ (2) = (2 + 3) / 5
f⁻¹ (2) = 5 / 5
f⁻¹ (2) = 1
2. Find the value of f⁻¹ (g(2)) as;
⇒ f⁻¹ (g(2)) = f⁻¹ (2 + 2)
= f⁻¹ (4)
= (4 + 3) / 5
= 7 / 5
Thus, f⁻¹ (g(2)) = 7 / 5.
3. Find the value of f (g⁻¹(2)) as;
The value of g⁻¹ (x) as;
g (x) = (2x + 4) / 2
g (x) = x + 2
Substitute g (x) = y and solve for x as;
g (x) = x + 2
y = x + 2
x = y - 2
Substitute x = g⁻¹ (x);
g⁻¹ (x) = x - 2
So, g⁻¹ (2) = 2 - 2 = 0
Hence,
⇒ f (g⁻¹(2)) = f (0)
= 5 × 0 - 3
= - 3
Thus, f (g⁻¹(2)) = - 3
Therefore, The solutions are;
1. f⁻¹ (2) = 1
2. f⁻¹ (g(2)) = 7 / 5
3. f (g⁻¹(2)) = - 3
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Hello, I need a bit of help with this question please.
Answers
To determine the initial length, we have to evaluate the given equation at x=0. Evaluating the equation, we get:
[tex]y=3\cdot0+59.[/tex]Therefore, the initial length of the road was 59 miles.
Now, notice that the given equation is a linear equation in slope-intercept form y=mx+b, where b is the slope, recall that the slope of a line represents the change of y compared to the change in x, in this case, miles per day.
Therefore, the change per day in the road's length is 3 miles.
Answer:
a) 59 miles.
b) 3 miles.
(2x + 15) Find the value of x? in the triangle
Answers
The given triangle is a right angle triangle. This means that one of the angles is 90 degrees. Recall, the sum of the angles in a triangle is 180 degrees. This means that
2x + 15 + x + 90 = 180
2x + x + 15 + 90 = 180
3x + 105 = 180
3x = 180 - 105
3x = 75
x = 75/3
x = 25
Find and solve a and b then verify the function
Answers
Hello!
We have the function f(x) = 3x.
The first step is to calculate the inverse function of f(x):First, let's replace where's f(x) by y:
f(x) = 3x
y = 3x
Now, let's swap the values of x and y:
y = 3x
x = 3y
Now we have to solve it to obtain y:
3y = x
y = x/3
So, we will have:
[tex]f(x)^{-1}=\frac{x}{3}[/tex]B. Reasoning:[tex]\begin{gathered} f(f^{-1}(x))=(3(\frac{x}{3})=x \\ \\ f^{-1}(f(x))=\frac{3x}{3}=x \end{gathered}[/tex]Image with the reasoning:
So, these equations are correct.
As it has no restrictions, this function is valid for all values of x.
Right answer: alternative A.
Obs: You'll have to type in the first box: x/3.
Determine the distance between the two points (-1,-9) and (4,-7)What is the midpoint of the line segment joining the pairs of Points.
Answers
The distance between two points (x₁,y₁) and (x₂,y₂) is given by the following formula.
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Then, we have:
[tex]\begin{gathered} (x_1,y_1)=(-1,-9) \\ (x_2,y_2)=(4,-7) \end{gathered}[/tex][tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=(4-(-1))^2+(-7-(-9))^2 \\ d=(4+1)^2+(-7+9)^2 \\ d=\sqrt{5^2+2^2} \\ d=\sqrt{25+4} \\ d=\sqrt{29} \\ d\approx5.4 \\ \text{ The symbol }\approx\text{ is read 'approximately'.} \end{gathered}[/tex]Finding the midpoint of the line segment joining the pointsThe midpoint of the line segment P(x₁,y₁) to Q(x₂,y₂) is:
[tex]\text{ Midpoint }=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Then, we have:
[tex]\begin{gathered} \text{ Midpoint }=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \text{ Midpoint }=(\frac{-1+4}{2},\frac{-9+(-7)}{2}) \\ \text{ Midpoint }=(\frac{3}{2},\frac{-16}{2}) \\ \text{ Midpoint }=(\frac{3}{2},-8) \end{gathered}[/tex]AnswerThe distance between the given points is √29 units or 5.4 units rounded to the nearest tenth.
The midpoint of the line segment that joins the pairs of points is (3/2,-8).
Solve for the system of equations by graphing. find the point of intersection.y = x+4y = -x+5 See picture of problems I need help with
Answers
Step 1
Given;
Step 2
We will graph for the solution using the points below.
[tex]\begin{gathered} For\text{ y=x+4} \\ (-4,0),(0.5,4.5)\text{ and \lparen0,4\rparen} \\ For\text{ y=-x+5} \\ (0,5),(0.5,4.5)\text{ and \lparen5,0\rparen} \end{gathered}[/tex]Answer; The solution and point of intersection is
[tex](0.5,4.5)[/tex]daniel picked 7 pounds of strawberry he wants to share the strawberry equally among three of his friends how many pounds of strawberries will each friend receive ?
Answers
Diana, this is the solution to the problem:
• Amount of strawberry Daniel picked = 7 pounds
,• Number of friends = 3
• Amount of strawberry that each friend will receive = Amount of strawberry Daniel picked/Number of friends
Replacing by the values we know:
• Amount of strawberry that each friend will receive = 7/3
,• Amount of strawberry that each friend will receive = 2.33 pounds
Btw the explanation is in words (example: because ? And ? are alternate interior angles, ? Is congruent to ?)
Answers
We know that ∠1 ≅ ∠4
As you determined ∠2 is the supplement of ∠1 and ∠3 is the supplement of ∠4, since ∠1 and ∠4 are equal, you can conclude that ∠2 and ∠3 are also equal.
Point R is the midpoint of XY, which indicates that it divides the line into two equal segments XR and RY.
Segment XR is formed by segments XP and PR, following the segment addition postulate you can determine that:
[tex]XR=XP+PR[/tex]Segment RY is formed by segments RT and TY, so that:
[tex]RY=RT+TY[/tex]Following the substitution postulate, if XR and RY are equal, then:
[tex]XP+PR=RT+TY[/tex]We know that XP ≅ TY, sp they can be simplified from the sums and we get that:
[tex]PR=RT[/tex]Finally, ∠QRP and ∠SRT are at opposite sides of the X shape formed by the intersection of lines QS and PT, they share a vertex at point R. These angles are vertically opposite angles and therefore congruent, so that:
[tex]\angle\text{QRP}\cong\angle SRT[/tex]You can conclude that ΔPQR and ΔTSR are congruent by the Angle-Side-Angle postulate.
Question number 12. Find the area of each sector.DO NOT ROUND.
Answers
For solving this problem we need to remember the generic formula. If we have a circle sector with angle x (in degrees),
[tex]\text{Area of S}=(radius)^2\cdot\pi\cdot(\frac{angle}{360})[/tex]The trick of this exercise is that our angle is expressed in degrees (°). Be careful!
Let's compute the solution:
[tex]Area\text{ of our sector }=(16\cdot\pi)^2\cdot\pi\cdot(\frac{240}{360})=16^2\cdot\pi^2\cdot\pi\cdot(\frac{2}{3})=\pi^3\cdot(\frac{512}{3})=\frac{512}{3}\cdot\pi^3[/tex]That's the final answer.
Comment: For every exercise of this kind you only need to apply the formula I provided you above. If the angle is in radians, the formula is
[tex]\text{ Area of sector }=\frac{1}{2}(radius)^2\cdot(angle)[/tex]PLEASE WILL GIVE BRAINIEST IF RIGHT THIS IS DUE TODAY PLEASE PLEASE
Can a triangle be formed with side lengths 15, 7, and 6? Explain.
No, because 7 + 6 < 15
No, because 6 + 7 > 15
Yes, because 15 + 7 > 6
Yes, because 15 + 6 < 7
Answers
Answer:
no because 7+6<15
Step-by-step explanation:
Answer:the first one a
Step-by-step explanation:
James has already saved $68 for the $200 video game system.
If he earns $5.50 per hour in his job, how many hours does he need to work to earn the
remainder of the money he needs to buy the video game system?
Answers
Answer: 24 hours
Step-by-step explanation:
First, we will find what he needs to make still.
$200 - $68 = $132
Next, we will divide by $5.50 to find the hours needed since he makes $5.50 per hour.
$132 / $5.50 = 24 hours
#10 Round your answer to the nearest to two decimal points
Answers
Answer: $108.75
Given:
P = $5000
r = 8.7% = 0.087
t = 3 months
We will use the formula for the simple interest rate to solve for the interest penalty
[tex]I=\text{Prt}[/tex]Substitute the given values to the formula and we will get:
[tex]\begin{gathered} I=\text{Prt} \\ I=(5000)(0.087)(\frac{3}{12}) \\ I=108.75 \end{gathered}[/tex]Therefore, the interest penalty is $108.75
what are the solutions of the compound inequality 2d + 3 < -11 or 3d - 9 > 15
Answers
Answer:
d ≤ –7 or d > 8.
Step-by-step explanation:
Given : 2d + 3 ≤ –11 or 3d – 9 > 15.
To find : What are the solutions of the compound inequality .
Solution : We have given 2d + 3 ≤ –11 or 3d – 9 > 15.
For 2d + 3 ≤ –11
On subtracting both sides by 3
2d ≤ –11 - 3 .
2d ≤ –14.
On dividing both sides by 2 .
d ≤ –7.
For 3d – 9 > 15.
On adding both sides by 9.
3d > 15 + 9 .
3d > 24 .
On dividing both sides by 3 .
d > 8 .
So, A. d ≤ –7 or d > 8.
Therefore, A. d ≤ –7 or d > 8.
A pulley is turning at an angular velocity of 14.0 rad per second. How many revolutions is the pulley making each second? (Hint: one revolution equals 2 pi rad)
Answers
Answer:
7/π ≈ 2.23 revolutions per second
Step-by-step explanation:
You want the know the angular velocity in revolutions per second of a pulley turning at 14.0 radians per second.
Unit ConversionThe velocity in rad/s can be converted to rev/s using the conversion factor ...
1 rev = 2π rad
The angular velocity is ...
[tex]\dfrac{14\text{ rad}}{\text{s}}\times\dfrac{1\text{ rev}}{2\pi\text{ rad}}=\dfrac{14}{2\pi}\,\dfrac{\text{rev}}{\text{s}}=\boxed{\dfrac{7}{\pi}\text{ rev/s}\approx2.23\text{ rev/s}}[/tex]
if vectors u=4, v=6 and w=3 prove that(⃗ × ) × ⃗ = ⃗ × ( × ⃗ )(they are supposed to all have arrows over them but it’s not fully working)
Answers
Given:
[tex](\vec{u}\times\vec{v})\times\vec{w}=\vec{u}\times\vec{(v}\times\vec{w})[/tex]u has direction ( 1, 0,0)
v has directions (1, 1,0)
w has directions(1, -2,1)
[tex](0,0,1)\times\vec{w}\Rightarrow\begin{bmatrix}{i} & {j} & {k} \\ {0} & {0} & {1} \\ {1} & {-2} & {1}\end{bmatrix}\Rightarrow i(0--2)-j(0-1)+k(0)\Rightarrow(2,1,0)[/tex]So (2,1,0) is the left hand side. The cross product gives us a direction between two vectors or the coordiantes it pointing to.
The right hand side:
[tex]\vec{u}\times(\vec{v}\times\vec{w})\Rightarrow\vec{u}\times\begin{bmatrix}{i} & {j} & {k} \\ {1} & {1} & {0} \\ {1} & {-2} & {1}\end{bmatrix}\Rightarrow\vec{u}\times(i\begin{bmatrix}{1} & {0} \\ {-2} & {1}\end{bmatrix}-j\begin{bmatrix}{1} & {0} \\ {1} & {1}\end{bmatrix}+k\begin{bmatrix}{1} & {1} \\ {1} & {-2}\end{bmatrix})[/tex][tex]\vec{u}\times(i(1-0)-j(1-0)+k(-2-1))\Rightarrow\vec{u}\times(1,-1,-3)[/tex][tex]\vec{u}\times(1,-1,-3)\Rightarrow\begin{bmatrix}{i} & {j} & {k} \\ {1} & {0} & {0} \\ {1} & {-1} & {-3}\end{bmatrix}\Rightarrow i\begin{bmatrix}{0} & {0} \\ {-1} & {-3}\end{bmatrix}-j\begin{bmatrix}{1} & {0} \\ {1} & {-3}\end{bmatrix}+k\begin{bmatrix}{1} & {0} \\ {1} & {-1}\end{bmatrix}[/tex][tex]i(0-0)-j(-3-0)+k(-1-0)\Rightarrow(0,3,-1)[/tex]So using (u x v) x w = u x (v x w) on the left hand side we got (2,1,0) and the right hand side we got (0,3,-1)
Therefore we have (2,1,0) = (0,3,-1) which can't be possible.
Answer:
[tex](\vec{u}\times\vec{v})\times\vec{w}\ne\vec{u}\times(\vec{v}\times\vec{w})\text{ because \lparen2,1,0\rparen }\ne\text{ \lparen0,3,-1\rparen from the example used.}[/tex]
