Kevonta, this is the solution:
Time of departure : 11:49
Time of travel : 4:32
In consequence, the estimated arrival time if the train departs in the morning is:
11 + 4 = 15 hours
49 + 32 = 81 minutes
81 minutes = 1 hour + 21 minutes
4:21 pm
If the train departs at night, the estimated time of arrival is:
4:21 am
given the equation 7x + 3 = 7X - _______ , what's would go in the blank to make each of the following true:so the equation is true for no values of xso the equation is true for all values of xso the equation is true for only one value of x
Let k be the number in the blank, so that:
[tex]7x+3=7x-k[/tex]Substract 7x from both sides:
[tex]3=-k[/tex]These two equations are equivalent regardless the value of x. We can change the conclusions that we may obtain by choosing different values for k.
Then, the equation:
[tex]7x+3=7x-0[/tex]Is true for no values of x.
If we want the equation to be false regardless of the value of x, then set k so that -k is different from 3. For example, set k=0:
[tex]\begin{gathered} 3=-0 \\ \Rightarrow3=0 \end{gathered}[/tex]Since this is contradictory, then there are no values of x that make the equation true.
If we want the equation to be true for all values of x, then 3=-k must be an identity. Then, let k=-3:
[tex]\begin{gathered} 3=-(-3) \\ \Rightarrow3=3 \end{gathered}[/tex]Then, the equation:
[tex]7x+3=7x-(-3)[/tex]Is true for all values of x.
If we want the equation to be true for only one value of x, we have to bring back x into the equation 3=-k. So, we can take k=x. This way, we would have:
[tex]\begin{gathered} 7x+3=7x-x \\ \Rightarrow3=-x \\ \Rightarrow x=-3 \end{gathered}[/tex]Find an angle θ with 0∘<θ<360∘ that has the same:
Sine function value as 210∘
θ = _____ degrees
Cosine function value as 210∘
θ = ______ degrees
sin theta = -0.5 degrees
cos theta=-0.86 degrees
hope it helped
The point (5,4) is rotated 270 degrees clockwise, would the answer be (-4,5)?
The image of the point (5, 4) after being rotated 270 degrees clockwise around the origin is (- 4, 5).
How to determine the image of a point by rotation around the origin
In this problem we find the case of a point to be rotated by a rigid transformation, represented by a rotation around the origin. The transformation rule is defined by the following expression:
P'(x, y) = (x · cos θ - y · sin θ, x · sin θ + y · cos θ)
Where:
(x, y) - Coordinates of point P(x, y).θ - Angle of rotation (counterclockwise rotation is represented by positive values).P'(x, y) - Coordinates of the resulting point.If we know that P(x, y) = (5, 4) and θ = - 270°, then the coordinates of the image are, respectively:
P'(x, y) = (5 · cos (- 270°) - 4 · sin (- 270°), 5 · sin (- 270°) + 4 · cos (- 270°))
P'(x, y) = (- 4, 5)
The image of the point (5, 4) is (- 4, 5).
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The volume for a rectangular prism is given by the formula V = l · w · h, where l is the length of the prism, w is the width of the prism, and h is the height of the prism.
If the volume of a rectangular prism with a height of 8 inches is 200 cubic inches and the base of the prism is a square, then what is the width of the rectangular prism?
A.
5 inches
B.
12.5 inches
C.
25 inches
D.
40 inches
If the volume of a rectangular prism with a height of 8 inches is 200 cubic inches and the base of the prism is a square, then the width of the rectangular prism is 5 inches
The volume of the rectangular prism = 200 cubic inches
The height of the rectangular prism = 8 inches
We know the volume of the rectangular prism = l×w×h
Where ls if the length of the base
w is the width of the base
h is the height of the rectangular prism
Given that the base of the prism is a square, then the length of the base is equal to width of the base
Consider
l = w = x
Substitute the values in the equation of volume
x × x × 8 = 200
[tex]x^{2}[/tex] × 8 = 200
[tex]x^{2}[/tex] = 25
x = 5 inches
Hence, if the volume of a rectangular prism with a height of 8 inches is 200 cubic inches and the base of the prism is a square, then the width of the rectangular prism is 5 inches
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Consider the line with the equation: − 5 y − 5 x = 15 Give the equation of the line parallel to Line 1 which passes through ( 3 , − 10 ) : Give the equation of the line perpendicular to Line 1 which passes through ( 3 , − 10 ) :
The equations of the parallel and perpendicular lines are y = −x−7 and y = x−13, respectively.
The equation of the given line "Line 1" is :−5y−5x = 15Simplify the equation.y + x = −3Write the equation in the slope-intercept form.y = mx + cy = −x−3The slope of the line "Line 1" is −1.The parallel line will have the same slope and it passes through the point (3, −10).y = −x + c−10 = −3 + cc = −7The equation of the parallel line is y = −x−7.The slope of the perpendicular line will be the negative reciprocal of the slope of the given line.The slope of the perpendicular line is −1/(−1) = 1.y = x + cIt passes through (3, −10).−10 = 3 + cc = −13The equation of the perpendicular line is y = x−13.To learn more about lines, visit :
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√X²+5x+4/X²+8x+16 - X²-3x-4/X²-16
The expression given as [x²+ 5x + 4]/[x² + 8x + 16] - [x² - 3x - 4]/[x²-16] simplifies to 0
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to evaluate the expression?The expression is given as
[x²+ 5x + 4]/[x² + 8x + 16] - [x² - 3x - 4]/[x²-16]
Factorize the expression
So, we have
[x²+ 5x + 4]/[x² + 8x + 16] - [x² - 3x - 4]/[x²-16] = [(x + 4)(x + 1)]/[(x + 4)(x + 4)] - [(x - 4)(x + 1)]/[(x + 4)(x - 4)]
Simplify the common factors
So, we have
[x²+ 5x + 4]/[x² + 8x + 16] - [x² - 3x - 4]/[x²-16] = [(x + 1)]/[(x + 4)] - [(x + 1)]/[(x + 4)]
The terms of the above equation are the same
So, the result of subtracting one from the other is 0
This gives
[x²+ 5x + 4]/[x² + 8x + 16] - [x² - 3x - 4]/[x²-16] = 0
Hence, the value of the expression is 0
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According to the Oxnard College Student Success Committee report in the previous year, we believe that 22% of students struggle in their classes because they don't spend more than 8 hours studying independently outside of a 4-unit class. For this year, you would like to obtain a new sample to estimate the proportiton of all Oxnard students who struggle in their classes because they don't study enough outside of the classrooms. You would like to be 95% confident that your estimate is within 1.5% of the true population proportion. How large of a sample size is required? Do not round mid-calculation.n =
The required sample size, using the z-distribution, is given as follows:
n = 253.
What is a confidence interval of proportions?A confidence interval of proportions has the bounds given by the rule presented as follows:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which the variables used to calculated these bounds are listed as follows:
[tex]\pi[/tex] is the sample proportion, which is also the estimate of the parameter.z is the critical value.n is the sample size.The equation for the margin of error is given by:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
The confidence level is of 95%, hence the critical value z has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.
The estimate and the margin of error are given as follows:
[tex]\pi = 0.22, M = 0.015[/tex]
Hence the required sample size is calculated as follows:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.015 = 1.96\sqrt{\frac{0.015(0.985)}{n}}[/tex]
[tex]0.015\sqrt{n} = 1.96\sqrt{0.015(0.985)}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{0.015(0.985)}}{0.015}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{1.96\sqrt{0.015(0.985)}}{0.015}\right)^2[/tex]
n = 253. (rounding up, as a sample size of 252 would result in a margin of error slightly above 1.5%).
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solve step through stepx + 2y = 83x - 2y = 0
Add both the equations
[tex]\begin{gathered} x+2y=8 \\ 3x-2y=0 \\ \text{Add left hand side terms together and right hand side terms together.} \\ x+2y+3x-2y=8+0 \\ 4x=8 \\ x=\frac{8}{4}=2 \end{gathered}[/tex]Substitute 2 for x in x+2y =8 to find y
[tex]\begin{gathered} 2+2y=8 \\ 2y=8-2 \\ 2y=6 \\ y=\frac{6}{2}=3 \end{gathered}[/tex]The solutions to the equations are x=2 and y=3.
Find the equation of the line that passes through the given point and has the given slope. (Use x as your variable.)(4, −3), m = −2
General equation of line:
[tex]y=mx+c[/tex]Where,
[tex]\begin{gathered} m=\text{slope} \\ c=y-\text{intercept} \\ (x,y)=(4,-3) \end{gathered}[/tex]Slope of line is -2 then:
[tex]\begin{gathered} y=mx+c \\ y=-2x+c \end{gathered}[/tex][tex](x,y)=(4,-3)[/tex][tex]\begin{gathered} y=-2x+c \\ -3=-2(4)+c \\ -3=-8+c \\ 8-3=c \\ 5=c \end{gathered}[/tex][tex]\begin{gathered} y=mx+c \\ y=-2x+5 \end{gathered}[/tex]Equation of line is y=-2x+5
Marsha thinks that because 123 is greater than 68, 0.123 must be greater than 0.68. Show and explain why Marsha is incorrect.
Let's begin by listing out the information given to us:
[tex]\begin{gathered} 123=1\text{0}0 \\ \text{ 20} \\ \text{ 3} \\ 123=1\text{ hundred + 2 tens + 3 units} \\ 68\text{ = 6 (10)} \\ \text{ 8} \\ 68\text{ = 6 tens + 8 units} \end{gathered}[/tex]To know which is greater between 0.123 To know which i3& 0.68, let's express it in tenth, hundredth & thousandth
[tex]\begin{gathered} 0.123=3\text{ thousandth + 2 hundredth + 1 tenth + 0 unit} \\ 0.123=0.003+0.02+0.1+0 \\ 0.68=\text{ 6 hundredth + 8 tenth + 0 unit} \\ 0.68=0.60+0.08+0 \end{gathered}[/tex]0.123 has 3 decimal places, which means it is 123/1000
0.68 has 2 decimal places, which means it is 68/100
When you multiply by 100, we have:
0.123 * 100 = 12.3
0.68 * 100 = 68
It becomes very evident that 0.68 is greater than 0.123
Use Heron's Area Formula to find the area of the triangle. (Round your answer to two decimal places.) A = 81°, b = 76, c = 39
First, let's find the measure of the third side, using the law of cosine:
[tex]\begin{gathered} a^2=b^2+c^2-2bc\cdot\cos(A)\\ \\ a^2=76^2+39^2-2\cdot76\cdot39\cdot\cos81°\\ \\ a^2=5776+1521-927.34\\ \\ a^2=6.369.66\\ \\ a=79.81 \end{gathered}[/tex]Now, let's use Heron's formula to calculate the area:
[tex]\begin{gathered} p=\frac{a+b+c}{2}=97.405\\ \\ A=\sqrt{p(p-a)(p-b)(p-c)}\\ \\ A=1463.75 \end{gathered}[/tex]What’s 2,0000000 +3000,00000
In order to add these two numbers, we need to add the algarisms that are in the same position.
Since the first number has 7 zeros and the second one has 8 zeros, the numbers 2 and 3 will not be added, since they are in different positions (the number 3 has more "value", since it's in a higher position).
So, adding these two numbers, we have:
Therefore the result of this sum is 320,000,000 (three hundred and twenty million).
Please show and explain this please
Answer:
b
Step-by-step explanation:
The root at [tex]x=1[/tex] has a multiplicity of 1, and corresponds to a factor of [tex](x-1)[/tex].
The root at [tex]x=-2[/tex] has a multiplicity of 1, and corresponds to a factor of [tex](x+2)[/tex].
The root at [tex]x=3[/tex] has a multiplicity of 2, and corresponds to a factor of [tex](x-3)^2[/tex].
Given the function f(x)=x^2 and g(x)=(-x)^2-1 what transformations will occur if both functions are graphed on the same coordinate grid?
We are given the following two functions
[tex]\begin{gathered} f(x)=x^2 \\ g(x)=(-x)^2-1 \end{gathered}[/tex]Recall that the rule for reflection over the y-axis is given by
[tex]f(x)\rightarrow f(-x)[/tex]As you can see, the graph of g(x) will be a reflection over the y-axis of the graph f(x).
Recall that the rule for vertical translation (upward) is given by
[tex]f(x)\rightarrow f(x)-d[/tex]The above translation will shift the graph vertically upward by d units.
For the given case, d = 1
As you can see, the graph of g(x) will be a vertical translation of the graph f(x)
Therefore, we can conclude that the graph of g(x) will be a reflection over the y-axis and a vertical translation of the graph f(x).
1st option is the correct answer.
An arrow is launched upward with a velocity of 320 feet per second from the top of a 30-foot building. What is the maximum height
attained by the arrow?
Answer: Maximum height reached is 804.88 m
Step-by-step explanation:
At maximum height velocity is zero
We have equation of motion v² = u² + 2as
Initial velocity, u = 320 ft/s = 97.536 m/s
Final velocity, v = 0 m/s
Acceleration, a = -9.81 m/s²
Substituting
v² = u² + 2as
0² = 97.536² + 2 x -9.81 x s
s = 484.88 m
So the arrow further a height of 484.88 m
Total height = 320 + 484.88 = 804.88 m
Liam went into a movie theater and bought 9 bags of popcorn and 8 drinks,
costing a total of $97.50. Jacob went into the same movie theater and bought
5 bags of popcorn and 4 drinks, costing a total of $51.50. Determine the price
of each bag of popcorn and the price of each drink.
Each bag of popcorn costs $
and each drink costs $
Answer:5.50 & $6
Step-by-step explanation:
Each bag of popcorn is $5.50 and the drink is $6.
How to calculate the number of popcorn and drink?Liam went into a movie theater and bought 9 bags of popcorn and 8 drinks costing a total of $97.50. This will be:
9p + 8d = 97.50
Jacob went into the same movie theater and bought 5 bags of popcorn and 4 drinks, costing a total of $51.50. This will be:
5p + 4d = 51.50
where P = popcorn
d = drink
The equations will be:
9p + 8d = 97.50
5p + 4d = 51.50
Multiply equation i by 5
Multiply equation ii by 9
45p + 40d = 487.50
45p + 36d = 463.50
Subtract
4d = 24
Divide.
d = 24/4
d = 6
Drink = $6
Since 9p + 8d = 97.50
9p + 8(6) = 97.50
9p + 48 = 97.50
9p = 97.50 - 48
p = $5.50
Popcorn cost $5.50.
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Which equation best represents the graph above?
The equation is (x+1)²-4.
Here the graph is in the form parabola, where the parabola is basically a graph for a quadratic function which is a curve in such an order that a point on it is equidistant from a fixed point called the focus of the parabola, and a fixed line called the directrix of the parabola. The general formula for a parabola is: y = a(x-h)² + k , where (h,k) signifies the vertex whereas the standard equation of a parabola is y² = 4ax.
For the given graph the vertex is (h,k)=(-1,-4)
so the equation for the parabola is
=>y= a(x+1)^²-4 .......1
Since we can see the parabola intercept at the x-axis at point (1,0), putting those in the above equation :
=>0=a(1+1)^2-4
=>4=a(2)^2
=>a=1
So, substituting the value of an in equation 1, we get
=>y= (x+1)^2-4
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Which of the following is the correct equation for this function?
A. Y=-x²+3x - 4
B. Y= (x+1)(x-3)
c. Y= (x + 1)(x − 3)
D. Y + 1 = -(x − 3)²
Answer the following Formula:[tex]5 \times 5 \times 6 \times 8 - 6 \times 9 \times 524 \times 8 \times 6 + 9 - 725 \times 6[/tex]
we have the expression
[tex]5\times5\times6\times8-6\times9\times524\times8\times6+9-725\times6[/tex]we know that
Applying PEMDAS
P ----> Parentheses first
E -----> Exponents (Powers and Square Roots, etc.)
MD ----> Multiplication and Division (left-to-right)
AS ----> Addition and Subtraction (left-to-right)
solve Multiplication First
5x5x6x8=1,200
9x524x8x6=226,368
725x6=4,350
substitute
1,200-6x226,368+9-4,350
solve
6x226,368=1,358,208
1,200-1,358,208+9-4,350
Solve the addition and subtraction
1,200-1,358,208+9-4,350=-1,361,349
answer is-1,361,3496. The equations of two lines are 6x - y = 2 and 4x - y = -8. What is thevalue of y in the solution for this system of equations?Your answer
The given system of equations : 6x - y = 2 and 4x - y = -8.
6x - y = 2 ( 1 )
4x - y = -8 ( 2 )
for the value of y, use substitution method:
Solve the equation (1) for y :
6x - y = 2
y = 6x - 2
Susbtitute the value of y in the equation ( 2 )
4x - y = - 8
4x - (6x - 2) = - 8
4x - 6x + 2 = - 8
4x -6x = - 8 - 2
-2x = -10
Divide both side by ( -2)
-2x/(-2) = -10/(-2)
x = 5
Substitute x = 5 in the equation ( 1 )
6x - y = 2
6( 5 ) - y = 2
30 - y = 2
y = 30 - 2
y = 28
So, the value of y is 28
Answer : y = 28
Simplify the expression √(112x^10 y^13 ). Show your work. please help
To create the abbreviated expression, combine similar terms by adding them all together. If the expression be [tex]$\sqrt{112 x^{10} y^{13}}$[/tex] then expression exists [tex]$4 \sqrt{7} x^5 y^6 \sqrt{y}$[/tex].
What is meant by an expression?Solving a math problem is the same thing as simplifying an expression. You basically strive to write an expression as simply as you can when you simplify it. In the conclusion, there shouldn't be any more multiplying, dividing, adding, or removing to do.
Start by recognizing the like terms, or terms with the same variables and exponents, in algebraic formulas. Then, to create the abbreviated expression, combine similar terms by adding them all together.
Let the expression be [tex]$\sqrt{112 x^{10} y^{13}}$[/tex]
separating the roots of the expression, we get
[tex]$\sqrt{112 x^{10} y^{13}}=\sqrt{112} \sqrt{x^{10}} \sqrt{y^{13}}$[/tex]
simplifying the above equation, we get
[tex]$=\sqrt{112} \sqrt{x^{10}} \sqrt{y^{13}}$[/tex]
Simplify [tex]$\sqrt{x^{10}}= x^5$[/tex]
[tex]$=\sqrt{112} x^5 \sqrt{y^{13}}$[/tex]
Simplify [tex]$\sqrt{y^{13}}=y^6 \sqrt{y}$[/tex]
[tex]$=\sqrt{112} x^5 y^6 \sqrt{y}$[/tex]
[tex]$=\sqrt{112}=4 \sqrt{7}$[/tex]
[tex]$=4 \sqrt{7} x^5 y^6 \sqrt{y}$[/tex]
Therefore, the correct answer is [tex]$4 \sqrt{7} x^5 y^6 \sqrt{y}$[/tex].
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the midpoint between (42, 33) and (-2, -5)?
Answer: (40, 28)
Step-by-step explanation:
(42+(-2), 33+(-5))=
(42-2, 33-5)=(40, 28)
a = 1/2bh, solve for b
a = 1/2bh, solve for b
that means -----> isolate the variable b
so
[tex]\begin{gathered} a=\frac{1}{2}\cdot b\cdot h \\ \text{Multiply by 2 both sides} \\ 2a=b\cdot h \\ \text{Divide by h both sides} \\ b=\frac{2a}{h} \end{gathered}[/tex]therefore
the answer is
b=(2a)/hA Gallup poll conducted in November of 2011 asked the following question, "What would you
say is the most urgent health problem facing this country at the present time?" The choices
were access, cost, obesity, cancer, government interference, or the flu. The responses were
access (27%), cost (20%), obesity (14%), cancer (13%), government interference (3%), or the
flu (less than 0.5%).
The following is an excerpt from the Survey Methods section. "Results for this Gallup poll are based on telephone interviews conducted Nov. 3-6, 2011, with a random sample of 1,012
adults ages 18 and older, living in all 50 U.S. states and the District of Columbia. For results
based on a total sample of national adults, one can say with 95% confidence that the maximum margin of sampling error is ±4 percentage points."
Based on this poll, we are 95% confident that between_____% and ______% of U.S. adults feel that access to health care is the most urgent health-related problem.
(Enter numbers only. Do not include the %, e.g. enter 50 not 50%)
Based on this poll, we are 95% confident that between 23% (lower limit) and 31% (upper limit) of U.S. adults feel that access to health care is the most urgent health-related problem.
How do we determine the lower and upper limits for the confidence level?The lower limit is the lowest percentage of poll participants who choose access to health care as the most urgent health-related problem.
The upper limit is to the highest percentage of poll participants who choose access to health care as the most urgent health-related problem.
Using the lower and upper limits, the Gallup poll can confidently estimate the range of the poll participants who pin-pointed access to health care as the most urgent.
Mean responses who choose access = 27%
Margin of error = ±4
Lower Limit = µ - margin of error
= 27% - 4%
=23%
Upper Limit = µ + margin of error
= 27% + 4%
=31%
Thus, at a 95% confidence level, the Gallup poll can claim that 27% ±4% of poll participants rated access to health care as the most urgent issue.
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AllusRotate the triangle 90° counterclockwisearound the origin and enter the newcoordinates.B'( 11 )Enter theC(1,1number thatbelongs in theA' ([?],[ ]green boxA(1,-1)B(4,-2)C(2,-4)Enter
For a 90 degrees counterclockwise rotation of a point, (x, y) about the origin, the new position would be (- y, x)
Thus,
For A', it becomes (- - 1, 1) = (1, 1)
For B', it becomes (- - 2, 4) = (2, 4)
For C', it becomes (- - 4, 2) = (4, 2)
After mark spent $24 on snacks for the movies, He had $12 left. How much money did mark start with?
We will investigate how to determine the amount of money Mark started off with at the beginning off the day.
We will assume and declare a variable to Mark's bank balance at the beginning off the day:
[tex]P\text{ = Inital balance}[/tex]Then mark sets out for movies and gets himself snacks to enjoy along his movies. The total receipt charged for his excursion is:
[tex]E\text{( expenses ) = \$24}[/tex]After his day expenses he will be left with a closing balance for the day. The closing balance of the day is expressed as:
[tex]\text{Closing Balance = Initial Balance - Expenses}[/tex]We are given that mark was left with $12. This means:
[tex]\text{Closing Balance = \$12}[/tex]Using the expression above we can write:
[tex]12\text{ = P - 24}[/tex]We will solve the above expression for initial balance ( P ) as follows:
[tex]\begin{gathered} P\text{ = 12 }+\text{ 24} \\ P\text{ = \$36} \end{gathered}[/tex]Therefore, the answer is:
[tex]\text{\$36}[/tex]Please describe some of these new postulates and include a diagram on the whiteboard to help explain how they are applied.You have learned some new ways in this module to prove that triangles are similar Please describe some of these new postulates and include a diagram on the whiteboard to help explain how they are applied.
The Solution:
Required:
To use the three theorems of similarity:
There are 3 theorems for proving triangle similarity:
AA Theorem
SAS Theorem
SSS Theorem
SAS Theorem
What happens if we only have side measurements, and the angle measures for each triangle are unknown? If we can show that all three sides of one triangle are proportional to the three sides of another triangle, then it follows logically that the angle measurements must also be the same.
SSS Theorem
Or what if we can demonstrate that two pairs of sides of one triangle are proportional to two pairs of sides of another triangle, and their included angles are congruent?
AA Theorem
As we saw with the AA similarity postulate, it’s not necessary for us to check every single angle and side in order to tell if two triangles are similar. Thanks to the triangle sum theorem, all we have to show is that two angles of one triangle are congruent to two angles of another triangle to show similar triangles.
Write the standard form equation of an ellipse with foci(-3,1)and(-3,13) and eccentricity e=0.6.
Given:
[tex]\begin{gathered} foci:-3,1),and,(-3,13) \\ And,eccentricity,e=0.6 \end{gathered}[/tex]To Determine: The standard form equation of an ellipse
Solution
The prime factorization of $756$ is
\[756 = 2^2 \cdot 3^3 \cdot 7^1.\]Joelle multiplies $756$ by a positive integer so that the product is a perfect square. What is the smallest positive integer Joelle could have multiplied $756$ by?
The smallest positive integer Joelle could have multiplied 756 by
15876
This is further explained below.
What is a perfect square?Generally, A perfect square number is a number in mathematics that, when its square root is calculated, yields a natural number.
To solve this problem we can do:
[tex]\sqrt{756}[/tex]
By properties of roots
[tex]\begin{aligned}&\sqrt{756}=\sqrt{6\cdot126} \\&=\sqrt{6 * 6 * 21} \\\\ =\sqrt{6^2 * 21} \\&=6 \sqrt{21}\end{aligned}[/tex]
So, so that the multiplication of 756 by an integer becomes a perfect square, you have to multiply it by 21 to make $21^2$ and thus "eliminate" the root.
756 * 21=15876
In conclusion, You can verify that 15876is a perfect square since root (15876)=132 and 132 is a natural number
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I can't figure out how to do (i + j) x (i x j)for vector calc
In three dimensions, the cross product of two vectors is defined as shown below
[tex]\begin{gathered} \vec{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \\ \vec{B}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k} \\ \Rightarrow\vec{A}\times\vec{B}=\det (\begin{bmatrix}{\hat{i}} & {\hat{j}} & {\hat{k}} \\ {a_1} & {a_2} & {a_3} \\ {b_1} & {b_2} & {b_3}\end{bmatrix}) \end{gathered}[/tex]Then, solving the determinant
[tex]\Rightarrow\vec{A}\times\vec{B}=(a_2b_3-b_2a_3)\hat{i}+(b_1a_3+a_1b_3)\hat{j}+(a_1b_2-b_1a_2)\hat{k}[/tex]In our case,
[tex]\begin{gathered} (\hat{i}+\hat{j})=1\hat{i}+1\hat{j}+0\hat{k} \\ \text{and} \\ (\hat{i}\times\hat{j})=(1,0,0)\times(0,1,0)=(0)\hat{i}+(0)\hat{j}+(1-0)\hat{k}=\hat{k} \\ \Rightarrow(\hat{i}\times\hat{j})=\hat{k} \end{gathered}[/tex]Where we used the formula for AxB to calculate ixj.
Finally,
[tex]\begin{gathered} (\hat{i}+\hat{j})\times(\hat{i}\times\hat{j})=(1,1,0)\times(0,0,1) \\ =(1\cdot1-0\cdot0)\hat{i}+(0\cdot0-1\cdot1)\hat{j}+(1\cdot0-0\cdot1)\hat{k} \\ \Rightarrow(\hat{i}+\hat{j})\times(\hat{i}\times\hat{j})=1\hat{i}-1\hat{j} \\ \Rightarrow(\hat{i}+\hat{j})\times(\hat{i}\times\hat{j})=\hat{i}-\hat{j} \end{gathered}[/tex]Thus, (i+j)x(ixj)=i-j