laws of exponent : multiplication and power to a poweranswer and help me step by step
Answers
All the numbers multiply in the normal way, and the powers of a power need to be multiplied.
[tex]72c^4d^8e^{10}[/tex]Related Questions
CAN SOMEONE HELP WITH THIS QUESTION?✨
Answers
Step-by-step explanation:
as this is a right-angled triangle, we use Pythagoras to get also c :
c² = a² + b² = 2² + 7² = 4 + 49 = 53
c = sqrt(53)
we know, sine = opposite/Hypotenuse.
so,
sin(A) = 2/sqrt(53) = 0.274721128...
from the norm circle we know cosine is the other leg of the right-angled triangle :
cos(A) = 7/sqrt(53) = 0.961523948...
tan(A) = sin(A)/cos(A) = 2/7 = 0.285714286...
sec(A) = 1/cos(A) = sqrt(53)/7 = 1.040015698...
csc(A) = 1/sin(A) = sqrt(53)/2 = 3.640054945...
cot(A) = 1/tan(A) = cos(A)/sin(A) = 7/2 = 3.50
oh, and FYI :
A = 15.9453959...°
You randomly select one card from a 52-card deck. Find the probability of selecting a black eight or a black king.
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The theoretical probability is defined as the ratio of the number of favourable outcomes to the number of possible outcomes.
We have only two black eights on the deck and two black kings on the deck, therefore, the amount of favourable outcomes is equal to their sum, which is 4. The total amount of possible outcomes are the amount of cards, 52. The probability of selecting a black eight or a black king is:
[tex]P=\frac{4}{52}=\frac{1}{13}[/tex]A parallelogram has a height of 7 meters and an area of 35 square meters. what would the sum of the bases be for a trapezoid with the same height and area as the parallelogram?
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The height of trapezoid is h = 7 m.
The area of trapezoid is A = 35 m^2.
The formula for the area of trapezoid is,
[tex]A=\frac{a+b}{2}\cdot h[/tex]Here, a and b are length of bases of trapezoid.
Substitute the values in the formula to obtain the sum of bases of trapezoid.
[tex]\begin{gathered} 35=\frac{a+b}{2}\cdot7 \\ \frac{a+b}{2}=\frac{35}{7} \\ a+b=5\cdot2 \\ =10 \end{gathered}[/tex]So sum of bases of trapezoid is equal to 10 meters.
Solve the quadratic equation by completing the square.x ^ 2 - 18x + 70 = 0 First, choose the appropriate form and fill in the blanks with the correct numbers. Then, solve the equation. Round your answer to the nearest hundredth. If there is more than one solution, separate them with commas.
Answers
Answer:
Form:
[tex]\boxed{(x-9)^2=11}[/tex]Solution:
[tex]x=12.32,5.68[/tex]Explanation:
Step 1. The expression we have is:
[tex]x^2-18x+70=0[/tex]And we are required to find the appropriate form after completing the square, and then the solution or solutions to the equation.
Step 2. Compare the given equation with the general quadratic equation:
[tex]ax^2+bx+c=0[/tex]Our values for a, b, and c are:
[tex]\begin{gathered} a=1 \\ b=-18 \\ c=70 \end{gathered}[/tex]Step 3. Using the value of b, find the following expression:
[tex](\frac{b}{2})^2[/tex]The result is:
[tex](-\frac{18}{2})^2\longrightarrow(-9)^2[/tex]Step 4. Take the original equation
[tex]x^2-18x+70=0[/tex]Move the +70 as a -70 to the right-hand side:
[tex]x^2-18x=-70[/tex]And now add to both sides the expression found in step 3 for (b/2)^2:
[tex]x^2-18x+(-9)^2=-70+(-9)^2[/tex]Step 5. Factor the left-hand side of the equation as a perfect square binominal:
[tex]\begin{gathered} P\operatorname{erf}ect\text{ square binomial formula:} \\ (a\pm b)^2=a^2\pm2ab+b^2 \end{gathered}[/tex]Applying this to our expression:
[tex](x-9)^2=-70+(-9)^2[/tex]Step 6. Solve the operations on the right-hand side:
[tex]\begin{gathered} (x-9)^2=-70+81 \\ \downarrow\downarrow \\ \boxed{\mleft(x-9\mright)^2=11} \end{gathered}[/tex]The form is the equation is:
[tex]\boxed{(x-9)^2=11}[/tex]Step 7. To find the value or values of x, solve for x in the previous equation:
[tex]\begin{gathered} (x-9)^2=11 \\ \downarrow\downarrow \\ x-9^{}=\pm\sqrt[]{11} \\ \downarrow\downarrow \\ x^{}=\pm\sqrt[]{11}+9 \end{gathered}[/tex]Step 8. To find the two solutions we use the '+' and '-' signs separately:
[tex]\begin{gathered} x^{}=\sqrt[]{11}+9\longrightarrow x=3.3166+9=12.3166 \\ x^{}=-\sqrt[]{11}+9\longrightarrow x=-3.3166+9=5.6834 \end{gathered}[/tex]Rounding these values for x to the nearest hundredth (2 decimal places):
[tex]\begin{gathered} x=12.32 \\ x=5.68 \end{gathered}[/tex]Answer:
Form:
[tex]\boxed{(x-9)^2=11}[/tex]Solution:
[tex]x=12.32,5.68[/tex]on a coordinate plane triangle XYZ is rotated 90 degrees counterclockwise about the origin to form triangle XYZ which conclusion is always true
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Given a 90° rotation of triangle XYZ, every segment will rotate 90°, meaning that each segment will form a right angle with the corresponding transformed segment.
Thereby, the correct answer is:
[tex]\bar{YZ}\perp\bar{Y^{\prime}Z^{\prime}}[/tex]Answer B
A new song has gone viral on the Internet. The website hosting the song uses the function f(t)=500t^2 to represent the number of daily hits over time, where t is time in days. Use the function to predict the day on which the number of daily hits reaches 1,000,000. Show your work.
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We will have the following:
[tex]\begin{gathered} 1000000=500t^2\Rightarrow t^2=2000 \\ \\ \Rightarrow t=\sqrt{2000}\Rightarrow t=20\sqrt{5} \\ \\ \Rightarrow t=44.72135955... \end{gathered}[/tex]So, on day 44 it will reach 1 000 000.
Be sure to fully show the system of equations for each problem and the process used to solve the system.You are starting an office-cleaning service. You decide to charge both large and small offices. You charge $55 for a small office and $85 for a large office. You clean 14 offices and make $920. How many small offices and how many large offices did you clean?
Answers
Given:
The charge for a small office = $55.
The charge for a large office = $ 85.
The total number of offices = 14.
The total amount = $ 920.
Required:
We need to find a number of small offices and large offices.
Explanation:
Let x be the number of the small office and y be the number of the large office.
The equation of the total number of offices.
[tex]x+y=14[/tex][tex]x=14-y[/tex]The equation of the total amount.
[tex]55x+85y=920[/tex]Substitute x =14-y in the equation.
[tex]55(14-y)+85y=920[/tex][tex]55\times14-55y+85y=920[/tex][tex]770+30y=920[/tex]Subtract 770 from both sides of the equation.
[tex]770+30y-770=920-770[/tex][tex]30y=150[/tex]Divide both sides by 30.
[tex]\frac{30y}{30}=\frac{150}{30}[/tex][tex]y=5[/tex][tex]Substitute\text{ y=5 in the equation x=14-y.}[/tex][tex]x=14-5[/tex][tex]x=9[/tex]Final answer:
The equations are
[tex]x+y=14[/tex][tex]55x+85y=920[/tex]The number of small offices are 9.
The number of large offices are 5.
multiply and simplify 2/5 × -7/4
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Answer:
[tex]\frac{2}{5}\times\frac{-7}{4}=\frac{-7}{10}[/tex]Explanation:
Given the expression:
[tex]\frac{2}{5}\times\frac{-7}{4}[/tex]This is the same thing as
[tex]\frac{2\times(-7)}{5\times4}[/tex]evaluating this, we have
[tex]\frac{-14}{20}[/tex]Simplifying this, we have
[tex]\frac{-7}{10}[/tex]Describe the vertical asymptote (s) and hole (s) for the graph of y = (x+2) (x+4)/ (x+4) (x+1)
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Given:
[tex]y=\frac{(x+2)(x+4)}{(x+4)(x+1)}[/tex]Required:
We need tofnind the vertical asymptote(s) and hole (s) for the graph of the given function.
Explanation:
Vertical asymptotes can be found when the numerator of the function is equal to zero.
The numerator of the given function is (x+4)(x+1)
[tex](x+4)(x+1)=0[/tex][tex](x+4)=0\text{ or }(x+1)=0[/tex][tex]x=-4\text{ or x=-1}[/tex]The asymptote of the given function is either x =-4 or x =-1.
Recall that a hole exists on the graph of a rational function when both the numerator and denominator of the function are equal to zero.
The common factor of the given rational function
hich is equivalent to RootIndex 5 StartRoot 1,215 EndRoot Superscript x?
243x
1,215 Superscript one-fifth x
1,215 Superscript StartFraction 1 Over 5 x EndFraction
243 Superscript StartFraction 1 Over x EndFraction
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The given expression is equivalent to [tex]$(1215)^{x/5}[/tex].
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have the following equation -
[tex]$(\sqrt[5]{1215})^{x}[/tex]
For [tex]$\sqrt[a]{x} = x^{1/a}[/tex]
Using the rule, we can write -
[tex]$(\sqrt[5]{1215})^{x}[/tex] = [tex]$(1215)^{x/5}[/tex]
Therefore, the given expression is equivalent to [tex]$(1215)^{x/5}[/tex].
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The cube root of our varies inversely with the square of S which to equations model this relationship?
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The question states as follows;
"The cube root of r varies inversely with the square of s."
The general form of an inverse relationship is shown below;
[tex]y=\frac{k}{x}[/tex]Substituting the variables, we would now have;
[tex]\sqrt[3]{r}=\frac{k}{s^2}[/tex]Therefore, the third option is correct.
Also;
[tex]\begin{gathered} \sqrt[3]{r}=\frac{k}{s^2} \\ \text{Observe that}_{} \\ \sqrt[3]{r}=r^{\frac{1}{3}} \end{gathered}[/tex]Therefore, we can alo have the expression;
[tex]\begin{gathered} r^{\frac{1}{3}}=\frac{k}{s^2} \\ \text{Cross multiply, and we'll have;} \\ s^2r^{\frac{1}{3}}=k \end{gathered}[/tex]The fifth option is also correct.
ANSWER:
The third and fifth options are both correct models of the inverse relationship given.
What are inequalites, and a example of one.
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Inequalities are relationships like equalities, what makes it different from equalitites is that on both sides are different expressions that allows to comparate them without been equal, for this we have 4 types
≤ less or equal than
≥ greater or equal to
< less than
> greater than
some examples are:
[tex]\begin{gathered} 3x-3<50 \\ 5x-45>33x \\ x^2-15x<34 \end{gathered}[/tex]Another difference between inequalities and equalities is that in equalities we obtain 1,2 or 3 solutions accronding to the degree of the equation, in inequalities we can obtain infinite number of solutions.
The temperature at 4 p.m. one day was - 6° Celsius. By 11 p.m the temperature had risen 11 degrees. Find thetemperature at 11 pmThe temperature at 11 p.m was 0°c.
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Since the temperature rises by 11 degrees we need to add this to the original temperature, then:
[tex]-6+11=5[/tex]Therefore the temperature at 11 pm is 5° Celsius.
convert the following fraction to a decimal 3 15/16 a 2,5472 b. 3.156 c. 3.0375 d. 4. 238
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ANSWER:
3.9375
EXPLANATION:
Given:
[tex]3\frac{15}{16}[/tex]To convert this to decimal, first convert the fraction, 15/16 to decimal:
[tex]\frac{15}{16}=\text{ 0.9375}[/tex]Now add the 3 whole number to 0.9375:
[tex]3\text{ + 0.9375 = 3.9375}[/tex]How many kg of water are in a 132 kg person ?
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Answer:
[tex]88kg\text{ of water}[/tex]Explanation:
Here, we want to get the number of kg of water in a 132 kg person
From the question, there is a direct variation between kg of water in the human body and the mass of the body
Let the mass be m and the amount of water be w
By direct variation, with k as the proportionality constant, we have it that:
[tex]w\text{ = }km[/tex]From the question, a 72 kg mass boy has 48 kg of water
Mathematically, we have it that:
[tex]\begin{gathered} 48\text{ = k }\times72 \\ k\text{ = }\frac{48}{72} \\ \end{gathered}[/tex]Now, we want to get the kg of water in a 132 kg person
What we have to do here is to have k as 48/72 and solve for w
Mathematically, we have that as:
[tex]\begin{gathered} w\text{ = }\frac{48}{72}\times132 \\ \\ w\text{ = 88 }kg \end{gathered}[/tex]Calculate the area of the circle shown below.10 in Approximate Value_________Exact Value________(round your approximate answers to thehundredths)Circumference of the circle:_________ in?_________ in?
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The radius of the circle is r=10 in.
The circumference of the circle is,
[tex]\begin{gathered} C=2\pi r \\ =2\pi\times10 \\ =20\pi \\ =62.83in \end{gathered}[/tex]Thus, the exact value of circumference is 20pi inches and the approximate value is 62.83 in.
Solve for v-2v-5v-17=25 Simplify your answer as much as possible
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Given:
-2v-5v-17=25
To determine the value of v, we first add similar elements first:
[tex]\begin{gathered} -2v-5v-17=25 \\ -7v-17=25 \end{gathered}[/tex]Next, we add 17 to both sides:
[tex]\begin{gathered} -7v-17+17=25+17 \\ Simplify \\ -7v=42 \\ \frac{-7v}{-7}=\frac{42}{-7} \\ v=-6 \end{gathered}[/tex]Therefore, the value of v is -6.
determine whether the given below each equation represents a direct variation or not if it does find the constant of the variation 3y = 4x
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,y= kx
Where k is the constant of variation.
First solve for y:
3y=4x
y= 4/3 x
So, the constant of variation k= 4/3
It represents a direct variation.
can you help pls with
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Answer:
C. 3.5=3.50
Explanation:
Given the statements below:
[tex]\begin{gathered} A.$0.43>0.5$ \\ B.$0.65<0.56$ \\ C.3.5=3.50 \\ D$.2.45>2.54$ \\ E.$0.4<0.04$ \end{gathered}[/tex]The only true statement out of the given options is:
[tex]3.5=3.50[/tex]The correct choice is C.
A researcher studying public opinion of proposed social security changes obtains a simple random sample of 50 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many more adult Americans does the researcher need to sample in the following cases? b. 25% of all adult Americans support the changes. b. The researcher must ask [] more American adults.
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b) We have 25% support the changes, therefore solve:
[tex]np(1-p)=10[/tex]where p =25% = 0.25
So
[tex]\begin{gathered} n(0.25)(1-0.25)=10 \\ n(0.25)(0.75)=10 \\ n(0.1875)=10 \\ \frac{n(0.1875)}{0.1875}=\frac{10}{0.1875} \\ n=53.3\approx54 \end{gathered}[/tex]Need 54 people but already have 50, then:
[tex]54-50=4[/tex]Answer: 4
Patrick has a swimming pool that needs to be drained. His pool holds9,644.6 gallons of water and will need to drain completely in 8 hours.What is the change in the water level per hour for Patrick's swimmingpool? Round your answer to the nearest hundredth.
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Explanation
We are asked to find the change in the water level per hour for Patrick's swimming
We have to use the formula
[tex]change\text{ in water level per hour=}\frac{Volume\text{ of water}}{Time\text{ taken to drain}}[/tex]Thus
[tex]change\text{ in water level per hour=}\frac{9644.6}{8}=1205.575\text{ gallons per hour}[/tex]Note: The rate will be negative because we are draining
Therefore, the change in water level will be -1205.58 gallons per hour
Two linear functions are shown below. Compare each fuoction to answer the questions. Function 2: Function 1: -11 8 -7 13 3 Ng -3 18 Part A: What is the rate of change for Function 1? Part B: What is the rate of change for Function 2? Part C: Which function has the greater rate of change?
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The rate of change of a linear functions is given by:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where (x1,y1) and (x2,y2) are points through the graph.
Function 1.
From the table we have that the functions passes through the points (-11,8) and (-7,13), pluggin the values in the formula above we have:
[tex]\begin{gathered} m=\frac{13-8}{-7-(-11)} \\ m=\frac{5}{11-7} \\ m=\frac{5}{4} \end{gathered}[/tex]Therefore the rate of change of functions 1 is 5/4
Function 2.
From the graph we notice that the functions passes through the points (-3,-4) and (1,-1), hence:
[tex]\begin{gathered} m=\frac{-1-(-4)}{1-(-3)} \\ m=\frac{-1+4}{1+3} \\ m=\frac{3}{4} \end{gathered}[/tex]Therefore the rate of change of function 2 is 3/4.
Comparing both rates of change we conclude that Function 1 has the greater change of rate.
What is the probability that a family with five children will have at least one boy? Write your answer as a percent rounded to the nearest whole.
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The answer is 0.96875.
Solution;
A family has five children
The probability that at least one of them is a boy = 1-P (all of them are girls)
= 1-(1/2)5
= 1-1/32
= 31/32
= 0.96875
Probability is simply the chance that something will happen. Whenever the outcome of an event is uncertain, we can speak of the probability, or likelihood, of a particular outcome. Analyzing events according to their probabilities is called statistics.
A probability sentence is a declarative sentence in which the term probability or one of its derivatives occurs. The modern mathematical theory of probability has its roots in the gambling experiments of his Gerolamo Cardano in 1654, Blaise Pascal and Pierre de Fermat laid the basic foundations of probability theory, making them the fathers of probability.
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please help me solve. The answer I have is incorrect which is 527.5
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527.8 m³
1) Let's find out the Volume of the biggest cylinder then subtract from it the smaller one.
Big Cylinder Volume:
[tex]\begin{gathered} V=\pi\cdot r^2\cdot h \\ V=\pi\cdot9^2\cdot3 \\ V=243\pi\approx763.4m^3 \end{gathered}[/tex]The Small one:
[tex]\begin{gathered} V=\pi\cdot5^2\cdot3 \\ V=75\pi m^3 \end{gathered}[/tex]Subtracting 75π from 243π we have:
V=243π -75π =168π approximately 527.7875658 to the nearest tenth: 527.8 m³
Find the slope of the linear function f with f(2) = 16 and f(4) = -2
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f(2) = 16, Let this be represented as (2, 16)
f(4) = -2, Let this be represented as (4, -2)
[tex]\begin{gathered} \text{slope =}\frac{change\text{ in y}}{\text{change in x}} \\ \\ \text{slope = }\frac{-2-16}{4-2} \\ \text{slope = }\frac{-18}{2} \\ \\ \text{slope}=\text{ -9} \end{gathered}[/tex]What is a multiple root of a polynomial and how do you find it?
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The Solution:
The given polynomial is
[tex]P(x)=2x^4-4x^3-16x^2[/tex]A root of the polynomial P(x) is the value of x for which the polynomial P(x) is equal to zero.
That is, any value of x that makes P(x) = 0, is a root of P(x).
The Multiplicity of a Root: This is the number of times a particular root appears as a root in a polynomial.
To find the root of a polynomial, say P(x), you have to equate P(x) to zero, and then solve for the value of x.
So, we shall follow the above procedures to find the root(s) of P(x), and thereafter determine if there are multiple roots.
[tex]\begin{gathered} P(x)=2x^4-4x^3-16x^2=0 \\ \text{Factoring out 2x}^2,\text{ we have} \\ 2x^2(x^2-2x-8)=0 \end{gathered}[/tex]This means that:
[tex]\begin{gathered} x^2-2x-8=0 \\ or \\ 2x^2=0 \end{gathered}[/tex]Solving quadratic equations above by Tthe Factorization Method, we get
[tex]\begin{gathered} x^2-2x-8=0 \\ x^2-4x+2x-8=0 \\ x(x-4)+2(x-4)=0 \\ (x-4)(x+2)=0 \end{gathered}[/tex]So,
[tex]\begin{gathered} P(x)=2x^2(x-4)(x+2)=0 \\ \text{This means} \\ 2x^2=0\text{ }\Rightarrow x=0 \\ x-4=0\text{ }\Rightarrow x=4 \\ x+2=0\text{ }\Rightarrow x=-2 \\ So,\text{ the roots of P(x) are 0, -2, and 4} \end{gathered}[/tex]Looking at the roots of P(x) above, there is no root that appears more than once, hence, the multiplicity of each of the roots is one.
put the following functions in order for smallest maxium to largest maxium
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Solve the following system of linear equations using elimination. -3x + 3y=-3 2x-y=0
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The system of equations is
[tex]\begin{gathered} -3x+3y=-3\Rightarrow(1) \\ 2x-y=0\Rightarrow(2) \end{gathered}[/tex]Since all terms in equation 1 can divide by 3, then
Divide each term in equation 1 by 3
[tex]\begin{gathered} \frac{-3x}{3}+\frac{3y}{3}=\frac{-3}{3} \\ -x+y=-1\Rightarrow(3) \end{gathered}[/tex]Add equations (2) and (3) to eliminate y
[tex]\begin{gathered} (2x-x)+(-y+y)=(0-1) \\ x+0=-1 \\ x=-1 \end{gathered}[/tex]Substitute x by -1 in equation (2) to find y
[tex]\begin{gathered} 2(-1)-y=0 \\ -2-y=0 \end{gathered}[/tex]Add y to both sides
[tex]\begin{gathered} -2-y+y=0+y \\ -2+0=y \\ -2=y \\ y=-2 \end{gathered}[/tex]The solution of the given system of equations is (-1, -2)
State if the three numbers can be the measures of the ustedes of a triangle. 12, 18, 9
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State if the three numbers can be the measures of the ustedes of a triangle.
12, 18, 9
Remember that
The triangle inequality theorem, states that
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
so
12+9 > 18
21 > 18 -----> is true
You only need to see that the two smaller sides are greater than the largest side
therefore
the answer is
Yes, the three numbers can be the measures of the ustedes of a triangleChoose the correct way to end the sentence.The lines x – 2y = 4 and y = 2x – 2 areA. parallelB. neitherC. perpendicular
Answers
Given the equations of the lines:
[tex]\begin{gathered} x-2y=4\rightarrow(1) \\ y=2x-2\rightarrow(2) \end{gathered}[/tex]We will write both equations in slope-intercept form to find the slope of each line:
The equation of the first line:
[tex]\begin{gathered} x-2y=4 \\ -2y=-x+4\rightarrow(\div-2) \\ \\ y=\frac{1}{2}x-2 \end{gathered}[/tex]so, the slope of the line (1) = 1/2
the equation of the second line:
[tex]y=2x-2[/tex]so, the slope of the second line = 2
Comparing the slopes of the lines:
1) the slopes are not equal, so the lines are not parallel
2) the product of the slopes = 1/2 * 2 = 1
So, the lines are not perpendicular
so, the answer will be option B. neither