Answer:
a) 3
b) 364
Step-by-step explanation:
A geometric sequence in explicit form is [tex]a_n=a_1 \cdot r^{n-1}[/tex] where [tex]a_1[/tex] is the first term and [tex]r[/tex] is the common ratio.
We are given:
[tex]a_5=9 \cdot a_3[/tex]
[tex]a_6+a_7=972[/tex].
What is a) r?
What is b) the sum of the first 6 terms?
So I'm going to use my first equation and use my explicit form to find those terms in terms of r:
[tex]a_1 \cdot r^4=9 \cdot a_1 \cdot r^{2}[/tex]
Divide both sides by [tex]a_1r^2[/tex]:
[tex]r^2=9[/tex]
[tex]r=\sqrt{9}[/tex]
[tex]r=3[/tex].
So part a is 3.
Now for part b).
We want to find [tex]a_1+a_2+a_3+a_4+a_5+a_6[/tex].
So far we have:
[tex]a_1=a_1[/tex]
[tex]a_2=3a_1[/tex]
[tex]a_3=3^2a_1[/tex]
[tex]a_4=3^3a_1[/tex]
[tex]a_5=3^4a_1[/tex]
[tex]a_6=3^5a_1[/tex].
We also haven't used:
[tex]a_6+a_7=972[/tex].
I'm going to find these terms in terms of r (r=3).
[tex]3^5a_1+3^6a_1=972[/tex]
[tex]243a_1+729a_1=972[/tex]
You have like terms to add:
[tex]972a_1=972[/tex]
Divide both sides by 972:
[tex]a_1=1[/tex]
The first term is 1 and the common ratio is 3.
The terms we wrote can be simplify using a substitution for the first term as 1:
[tex]a_1=a_1=1[/tex]
[tex]a_2=3a_1=3(1)=3[/tex]
[tex]a_3=3^2a_1=9(1)=9[/tex]
[tex]a_4=3^3a_1=27(1)=27[/tex]
[tex]a_5=3^4a_1=81(1)=81[/tex]
[tex]a_6=3^5a_1=243(1)=243[/tex].
Now we just need to find the sum of those six terms:
1+3+9+27+81+243=364.
I need this factored. Is it actually considered prime??
Given 3 non-collinear points, which of the following are true?
The intersection of 2 planes would contain all 3 points.
They will be contained in the same line.
There is only 1 plane that contains all 3 points.
Only one line can be drawn containing any 2 of the points.
Answer:
There is only 1 plane that contains all 3 points.
Step-by-step explanation:
According to the three point postulate, three non-collinear points are in one plane. Therefore, your answer would be there is only 1 plane that contains all 3 points.
Let's analyze each statement regarding the three non-collinear points:
1. "The intersection of 2 planes would contain all 3 points."
This statement is false. The intersection of two planes in three-dimensional space is a line, and there is no guarantee that a line resulting from the intersection of two arbitrary planes will contain all three non-collinear points. In fact, the likelihood of this happening by chance is zero.
2. "They will be contained in the same line."
This statement is false as well. Since the points are non-collinear, by definition, they do not all lie on a single line. A line can only contain two of the points at a time, but not all three if they are non-collinear.
3. "There is only 1 plane that contains all 3 points."
This statement is true. Given any three non-collinear points in space, there is exactly one plane that contains all three. This is because any three points that are not on the same line can define a plane by being unique points in a two-dimensional subspace of three-dimensional space.
4. "Only one line can be drawn containing any 2 of the points."
This statement is true as well. For any two distinct points, there exists exactly one line that connects them. This is one of the fundamental principles of geometry: through any two points, there is exactly one straight line.
In summary, the third and fourth statements are true, while the first and second are false.
The eucalyptus is the world's fastest growing tree. It grows an average of 2 1/2 centimeters every day. If a eucalyptus tree is 50 centimeters tall when it is planted, how tall will it be in 5 days?
a piece of rope 11/12 yd long is cut into two pieces. One piece is 4/7 yd long. How long is the other piece?
Answer:
(4/7) + x = (11/12)
(11/12) -(4/7) = x
We need to convert BOTH denominators to 84
(11/12) * 7 = 77 / 84
(4 / 7) * 12 = 48 / 84
77 / 84 -(48 / 84) = 29 / 84
Step-by-step explanation:
Your report card contains five A's and three B's. What is the ratio of A's to B's?
Answer:
5 : 3
Step-by-step explanation:
Five A's
Three B's
Therefore, 5 : 3
You would not include the letters because a ratio is just a number.
Find the perimeter of a parallelogram if two of its adjacent sides are 25 inches and 30 inches.
Answer:
P=110in
Step-by-step explanation:let me know if it's correct i'm not 100% sure
Answer:
110 inches
Step-by-step explanation:
The perimeter is basically the sum of all sides
In this case, it will be
25+25=50 for 2 opposite sides
30+30= 60 for 2 other opposite sides
hence 50+60= 110 inches
URGENT!!!!!Driving times for students' commute to school is normally distributed, with a mean time of 14 minutes and a standard deviation of 3 minutes. Using the empirical rule, approximately what percent of students' commute time is between 11 and 17 minutes? 32% 68% 95% 99.7%
Answer:
B. 68%.
Step-by-step explanation:
We have been given that driving times for students' commute to school is normally distributed, with a mean time of 14 minutes and a standard deviation of 3 minutes.
First of all, we will find z-score of 11 and 17 using z-score formula.
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{11-14}{3}[/tex]
[tex]z=\frac{-3}{3}[/tex]
[tex]z=-1[/tex]
[tex]z=\frac{17-14}{3}[/tex]
[tex]z=\frac{3}{3}[/tex]
[tex]z=1[/tex]
We know that z-score tells us a data point is how many standard deviations above or below mean.
Our z-score -1 and 1 represent that 11 and 17 lie within one standard deviation of the mean.
By empirical rule 68% data lies with in one standard deviation of the mean, therefore, option B is the correct choice.
Answer: 68%
Step-by-step explanation: ya boy just took le test :-)
0.7 of 12.99
how do you solve it?
Answer:
9.093
Step-by-step explanation:
Of means multiply
.7 * 12.99
9.093
Answer:
9,093
Step-by-step explanation:
Yes. You take 70% of 12,99 [multiply].
I am joyous to assist you anytime.
8c - c +6=48. How do I explain this with words?
I Need Help Failing Badly Geometry Is Hard!!
Answer:
Choice A. Segment LM is congruent to segment LO.
Step-by-step explanation:
Triangles LMX and LOX are right triangles since we see that each one has a right angle.
Segment LX is congruent to itself. Segment LX is a side of both triangles. It is a leg of both triangles, so we already have a leg of one triangle congruent to a leg of the other triangle.
For the HL theorem to work, we need a leg and the hypotenuse of one triangle to be congruent to the corresponding parts of the other triangle. Since we already have a pair of legs, we need a pair of hypotenuses.
The hypotenuses of the triangles are segments LM and LO.
Answer: A. Segment LM is congruent to segment LO.
PLEASE HELP URGENT!!!! what is the measure of angle C? 38 degrees. 76 degrees. 90 degrees. 152 degrees.
Answer:
38
Step-by-step explanation:
less than 45
Answer:38
Step-by-step explanation:
What is the area of parallelogram ABCD?
11 square units
13 square units
15 square units
16 square units
Answer:
13 square units
Step-by-step explanation:
First of all, you need to identify that ABCD is a rectangle (AB=CD and AD=BC).
The area of a rectangle is calculated by multiplying the length and the width.
Secondly, we use the Pythagoras’s theorem to calculate side CD and AD (the length and width). I’ve added some labels to your original diagram (see picture attached) so that it’s easier to understand.
The Pythagoras’s theorem is a^2 + b^2 = c^2 (c is the hypotenuse).
So, for side CD:
3^2 + 1^2 = (CD)^2
9 + 1 = (CD)^2
CD = √ 10
and for side AD:
4^2 + 1^2 = (AD)^2
16 + 1 = (AD)^2
AD = √17
Lastly, to calculate the area:
√10 x √17 = 13.04
Your answer is 13 square units.
Hope this helped :)
Answer:
Option B. 13 square units
Step-by-step explanation:
Area of a parallelogram is defined by the expression
A = [tex]\frac{1}{2}(\text{Sum of two parallel sides)}[/tex] × (Disatance between them)
Vertices of A, B, C and D are (3, 6), (6, 5), (5, 1) and (2, 2) respectively.
Length of AB = [tex]\sqrt{(x-x')^{2}+(y-y')^{2}}[/tex]
= [tex]\sqrt{(5-6)^{2}+(6-3)^{2}}[/tex]
= [tex]\sqrt{10}[/tex]
Since length of opposite sides of a parallelogram are equal therefore, length of CD will be same as [tex]\sqrt{10}[/tex]
Now we have to find the length of perpendicular drawn on side AB from point D or distance between parallel sides AB and CD.
Expression for the length of the perpendicular will be = [tex]\frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^{2}+B^{2}}}[/tex]
Slope of line AB (m) = [tex]\frac{y-y'}{x-x'}[/tex]
= [tex]\frac{6-5}{3-6}=-(\frac{1}{3} )[/tex]
Now equation of AB will be,
y - y' = m(x - x')
y - 6 = [tex]-\frac{1}{3}(x-3)[/tex]
3y - 18 = -(x - 3)
3y + x - 18 - 3 = 0
x + 3y - 21 = 0
Length of a perpendicular from D to side AB will be
= [tex]\frac{|(2+6-21)|}{\sqrt{1^{2}+3^{2}}}[/tex]
= [tex]\frac{13}{\sqrt{10}}[/tex]
Area of parallelogram ABCD = [tex]\frac{1}{2}(AB+CD)\times (\text{Distance between AB and CD})[/tex]
= [tex]\frac{1}{2}(\sqrt{10}+\sqrt{10})\times (\frac{13}{\sqrt{10} } )[/tex]
= [tex]\sqrt{10}\times \frac{13}{\sqrt{10} }[/tex]
= 13 square units
Option B. 13 units will be the answer.
Solve 2c – 8f = 24 for f. Show your work.
Answer: 3 - [tex]\frac{c}{4}[/tex]
Step-by-step explanation:
2c - 8f = 24
2(c - 4f) = 2(12)
c - 4f = 12
4f = 12 - c
F = 3 - [tex]\frac{c}{4}[/tex]
Why can’t a line or Ray have a perpendicular bisector
Answer:
Because they are both infinitely long.
Step-by-step explanation:
A ray goes on to infinity from a given point in one direction, whereas a line goes on to infinity in both directions.
Final answer:
A line or ray cannot have a perpendicular bisector because they extend infinitely without definite endpoints, thus lacking a midpoint for bisecting. Only a line segment, which has two endpoints, can have a perpendicular bisector that divides it into two equal parts at a right angle.
Explanation:
The question why a line or ray can't have a perpendicular bisector can be explained through geometric principles. A ray, by definition, is a line that starts at a point and extends infinitely in one direction. It doesn't have a midpoint or an end, and therefore cannot be bisected. Similarly, a line extends infinitely in both directions and does not have a midpoint for bisection. The concept of a perpendicular bisector requires a line segment, which has two endpoints, allowing for a midpoint to be determined and a line to be drawn at a 90-degree angle, equally dividing it into two equal parts.
Considering Euclidean geometry, it's understood that two perpendiculars cannot be parallel to the same line as they would then be parallel to each other, contradicting the definition of perpendicular lines. Moreover, a perpendicular bisector is defined in the context of a line segment within a plane, where the extremities of the segment are known, and there's a definite length to bisect.
Using Hyperbolic Geometry, it's also noted that if there were two common perpendiculars, a rectangle would form, which is not possible in that geometry. This further establishes the distinct properties between lines, rays, and line segments regarding the possibility of establishing perpendicular bisectors.
Find the area of the trapezoid.
Answer:
[tex]\large\boxed{A=54\ m^2}[/tex]
Step-by-step explanation:
The formula of an area of a trapezoid:
[tex]A=\dfrac{b_1+b_2}{2}\cdor h[/tex]
b₁, b₂ - bases
h - height
We must use the Pythagorean theorem:
[tex]x^2+8^2=10^2[/tex]
[tex]x^2+64=100[/tex] subtract 64 from both sides
[tex]x^2=36\to x=\sqrt{36}\\\\x=6\ m[/tex]
We have b₁ = 6 + 6 = 12m, b₂ = 6m and h = 8m.
Substitute:
[tex]A=\dfrac{12+6}{2}\cdot6=\dfrac{18}{2}\cdot6=(9)(6)=54\ m^2[/tex]
Someone help me answer this
Answer:
[tex]\large\boxed{1.\ V=\dfrac{80\pi}{3}\ cm^3\approx83.73\ cm^3}\\\boxed{2.\ V=\dfrac{28\pi}{3}\ cm^3\approx29.31\ cm^3}\\\boxed{3.\ V=36\pi\ in^3\approx113.04\ cm^3}[/tex]
Step-by-step explanation:
The formula of a volume of a cone:
[tex]V=\dfrac{1}{3}\pi r^2H[/tex]
r - radius
H - height
[tex]\pi\approx3.14[/tex]
[tex]\bold{1.}\\\\r=4cm,\ H=5cm\\\\V=\dfrac{1}{3}\pi(4^2)(5)=\dfrac{1}{3}\pi(16)(5)=\dfrac{80\pi}{3}\ cm^3\approx\dfrac{(80)(3.14)}{3}=83.73\ cm^3[/tex]
[tex]\bold{2.}\\\\r=2cm,\ H=7cm\\\\V=\dfrac{1}{3}\pi(2^2)(7)=\dfrac{1}{3}\pi(4)(7)=\dfrac{28\pi}{3}\ cm^3\approx\dfrac{(28)(3.14)}{3}=29.31\ cm^3[/tex]
[tex]\bold{3.}\\\\r=6in,\ H=3in\\\\V=\dfrac{1}{3}\pi(6^2)(3)=\dfrac{1}{3}\pi(36)(3)=36\pi\ in^3\approx(36)(3.14)=113.04\ in^3[/tex]
given the function f(x) =2x-5 and g(x) which function has a greater slope
x g(x)
2 0
4 5
6 10
A. f(x) has a greater slope
B. g(x) has a greater slope
C. the slopes of f(x) and g(x) are the same
D. the slope of g(x) is undefined
Answer:
B. g(x) has a greater slope.Step-by-step explanation:
The slope-intercept of an equation of a line:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
=========================================
[tex]f(x)=2x-5\to m=2[/tex]
From the table of function g(x) we have:
x = 2 → y = 0
x = 4 → y = 5
Calculate the slope:
[tex]m=\dfrac{5-0}{4-2}=\dfrac{5}{2}=2.5[/tex]
The slope of f(x) is equal to 2.
The slope of g(x) is equal to 2.5.
2 < 2.5
Answer:
B. g(x) has a greater slope
Step-by-step explanation:
Given the function f(x) =2x-5 and g(x), g(x) has a greater slope.
f(x) = 2
g(x) = 2.5
Henry, Brian and Colin share some sweets in the ratio 5:4:2. Henry gets 15 more sweets than Colin. How many sweets does Brian get?
Answer:
20 sweets.
Step-by-step-explanation:
Let Colin have x sweets.
The Henry gets x+15 sweets
Then according to the ratios:
5/2 = x+15/x
5x = 2x + 30
3x = 30
x = 10.
So Colin has 10 sweets.
The ratio of Brian's sweets to Colin's sweets is 4: 2 or 2:1.
So Brian has 2 * 10 = 20 sweets.
In 1995, the moose population in a park was measured to be 4200. By 1998, the population was measured again to be 1600. If the population continues to change linearly:
Find a formula for the moose population, P, in terms of t, the years since 1990.
P=
What does your model predict the moose population to be in 2003?
Answer:
P = -2600/3 t + 25600/3
P = -8200/3
Step-by-step explanation:
t is the time in years since 1990, so two points on the line are (5, 4200) and (8, 1600).
Using the points to find the slope:
m = (y₂ − y₁) / (x₂ − x₁)
m = (1600 − 4200) / (8 − 5)
m = -2600/3
Now writing the equation in point-slope form:
P − 4200 = -2600/3 (t − 5)
Converting to slope-intercept form:
P − 4200 = -2600/3 t + 13000/3
P = -2600/3 t + 25600/3
In 2003, t = 13:
P = -2600/3 (13) + 25600/3
P = -8200/3
The linear formula for the moose population is P = -800t + 8200. The moose population predicted by this model for the year 2003 is 2,400.
Explanation:In this question, we are given that in 1995 the moose population was 4200 and by 1998 it was 1600. This change in population mimics a linear relationship. We are asked to find the formula for this line and then predict the moose population in 2003.
We know that 1995 corresponds to t = 5 (since t is the years since 1990) and 1998 corresponds to t = 8. Therefore we can find the slope of the line (m) as (4200- 1600) / (5 - 8) = -800 per year. Since we know that the line crosses the point (5,4200), we can find the y-intercept, denoted as (b), using the formula y = mx + b.
Substitute m = -800, x = 5, and y = 4200 into the equation and solve for b
4200 = -800 * 5 + b
This simplifies to b= 4200 + 4000 = 8200
Therefore, the formula is P = -800t + 8200
To predict the moose population in 2003, simply substitute t = 13 into the formula (since 2003 is 13 years since 1990). Therefore,
P = -800 * 13 + 8200 = 2400
Therefore, the model predicts that the moose population in 2003 would be 2400.
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What number should be added to both sides of the equation to complete the squan
х2 + 8x = 4
оооо
Answer:
16
Step-by-step explanation:
х2 + 8x = 4
Take the coefficent of the x term, 8
Divide it by 2. 8/2 =4
Then square it, 4^2 = 16
Add this to both sides of the equation
x^2 +8x+16 = 4+16
Solve x^2-8x=3 by completing the square. Which is the solution set of the equation
Answer:
{-0.36, 8.36) to the nearest hundredth.
Step-by-step explanation:
x^2 - 8x = 3
(x - 4)^2 - 16 = 3
(x - 4)^2 = 19
Taking square roots:
x - 4 = +/- √19
x = 4 +/- √19
x = {-0.36, 8.36} to nearest 1/100.
For this case we have the following expression:
[tex]x ^ 2-8x = 3[/tex]
We must complete squares.
So:
We divide the middle term between two and we square it:
[tex](\frac {-8} {2}) ^ 2[/tex], then:
[tex]x ^ 2-8x + (\frac {-8} {2}) ^ 2 = 3 + (\frac {-8} {2}) ^ 2\\x ^ 2-8x + (- 4) ^ 2 = 3 + 16[/tex]
We have to, by definition:
[tex](a-b) ^ 2 = a ^ 2-2ab + b ^ 2[/tex]
Then, rewriting:
([tex](x-4) ^ 2 = 19[/tex]
To find the roots, we apply square root on both sides:
[tex]x-4 = \sqrt {19}[/tex]
We have two solutions:
[tex]x_ {1} = \sqrt {19} +4\\x_ {2} = - \sqrt {19} +4[/tex]
Answer:
([tex](x-4) ^ 2 = 19\\x_ {1} = \sqrt {19} +4\\x_ {2} = - \sqrt {19} +4[/tex]
is my working step wrong? the quaestion is find the range of values of x that satisfy the inequalities by using basic definition
Answer:
x < -2
Step-by-step explanation:
2|x| > 3x + 10
Divide both sides by 2.
|x| > 1.5x + 5
********************************************************
An absolute value inequality of the form
|X1| > X2
where X1 and X2 are expressions in x is solved by solving the compound inequality
X1 > X2 or X1 < -X2
********************************************************
Back to your problem.
|x| > 1.5x + 5
x > 1.5x + 5 or x < -(1.5x + 5)
-0.5x > 5 or x < -1.5x - 5
x < -10 or 2.5x < -5
x < -10 or x < -2
Since x < -10 is included in x < -2, the solution is
x < -2
What is the image of (-8,10) when reflected in the y-axis
Answer:
(8,10)
Step-by-step explanation:
If we reflect a point over the y-axis, the x becomes opposite while the y stays the same.
So the rule here is (a,b)->(-a,b) if we are reflecting over y-axis.
So if you reflect (-8,10) over the y-axis you get (8,10).
If f(x)=x2+3x+5, what is f(a+h)?
Answer:
[tex]\large\boxed{D.\ a^2+2ah+h^2+3a+3h+5}[/tex]
Step-by-step explanation:
[tex]f(x)=x^2+3x+5\\\\f(a+h)\to\text{exchange x to (a + h)}:\\\\f(a+h)=(a+h)^2+3(a+h)+5\\\\\text{use}\ (a+b)^2=a^2+2ab+b^2\ \text{and the distributive property}\\\\f(a+h)=a^2+2ah+h^2+3a+3h+5[/tex]
What is 5x times (3x^2 -5)
Answer:
[tex]\large\boxed{5x\times(3x^2-5)=15x^3-25x}[/tex]
Step-by-step explanation:
[tex]5x\times(3x^2-5)\qquad\text{use the distributive property:}\ a(b+c)=ab+ac\\\\=(5x)(3x^2)+(5x)(-5)\\\\=15x^3-25x[/tex]
The resulting product of the functions using the distributive property is
15x³ - 25x.
Product is an operation carried out when two or more variables, numbers, or functions are multiplied together.
Given the expression 5x(3x² - 5)
Taking the product:
5x(3x² - 5)
Expand using the distributive property
= 5x(3x²) - 5x(5)
= (5×3)(x × x²) - 25x
= 15x³ - 25x
Hence the resulting function is 15x³ - 25x.
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Could someone help me with this math problem?
Answer:
729
Step-by-step explanation:
1/3^-2×3^-4×(-1)^2
=3^2×3^4/1
=9×81
=729
For this case we have the following expression:
[tex]\frac {1} {3^ {- 2} * x^{ - 4} * y ^ 2}[/tex]
We must evaluate the expression to:
[tex]x = 3\\y = -1[/tex]
So:
[tex]\frac {1} {3^{- 2} * 3^{ - 4} * (- 1) ^ 2} =[/tex]
[tex]\frac {1} {\frac {1} {3 ^ 2} * \frac {1} {3 ^ 4} * 1} =\\\frac {1} {\frac {1} {3 ^ 2} * \frac {1} {3 ^ 4}} =\\\frac {1} {\frac {1} {9} * \frac {1} {81}} =\\\frac {1} {\frac {1} {729}} =\\\frac {729} {1} =\\729[/tex]
Answer:
Option B
I need these questions answered please
Answer:
Discontinuities are created when the denominator of the rational expression equals zero (because division by zero is undefined). Graphically, this is usually represented by a dashed vertical line indicating a vertical asymptote.
Which best describes the solutions to the inequality x>10?
(A)10 and every whole number greater than 10
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
(C)11 and every whole number greater than 11
(D)a rational number infinitely close to 10 but greater than 10, and all other whole numbers greater than 10
Answer:
It can be B or C (B is closer to being correct); neither are worded perfectly correct. Definitely not A or C.
Step-by-step explanation:
I am a math teacher and whoever created this question didn't cover everything.
The answer is all numbers greater than 10, not just rational (irrational should also be included) and not just whole numbers (fractions and decimals should be included).
Answer:
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
Step-by-step explanation:
x> 10 means all numbers greater than 10
(A)10 and every whole number greater than 10
False, does not include 10
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
True
(C)11 and every whole number greater than 11
False, it only included integers. 10.5 is a solution but not included here
(D)a rational number infinitely close to 10 but greater than 10, and all other whole numbers greater than 10
False, it only included whole numbers. 10.5 is a solution but not included here
The answer should really be real numbers, not rational numbers.
Irrational numbers can be solutions. But given the choices given, B is the best solution.
When Point E (-9, 3) is rotated 270° counterclockwise about the origin, it becomes Point E’ (3, -9). true or false?
Answer:
False
Step-by-step explanation:
It would be at -3,9.
what is the simplified form of sqaure root 72x to the power 16 over 50x 36 assume x = 0
1)6 over 5x power of 10
2)6 over 5x to power of 2
3)6 over 5x to the power of 10
4)6 over 5x to the power of 2
Answer:
[tex]\large\boxed{\dfrac{6}{5x^{10}}}[/tex]
Step-by-step explanation:
[tex]\sqrt{\dfrac{72x^{16}}{50x^{36}}}\qquad\text{simplify}\\\\=\sqrt{\dfrac{36x^{16}}{25x{^{20+16}}}}\qquad\text{use}\ (a^n)(a^m)=a^{n+m}\\\\=\sqrt{\dfrac{36x^{16}}{25x^{20}x^{16}}}\qquad\text{cancel}\ x^{16}\\\\=\sqrt{\dfrac{36}{25x^{20}}}\qquad\text{use}\ \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\ \text{and}\ \sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\\\\=\dfrac{\sqrt{36}}{\sqrt{25}\cdot\sqrt{x^{10\cdot2}}}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\dfrac{6}{5\sqrt{(x^{10})^2}}\qquad\text{use}\ \sqrt{a^2}=a\ \text{for}\ a\geq0\\\\=\dfrac{6}{5x^{10}}[/tex]