Applying the product rule for exponent numbers, 2⁴ . 2⁵ = 2⁴⁺⁵ = 2⁹
Recall the rules of exponent operations:
Product. [tex]a^{b}.a^{c}=a^{b+c}[/tex]Quotient. [tex]a^{b}:a^{c}=a^{b-c}[/tex]Power. [tex](a^{b})^{c}=a^{b.c}[/tex]Power of a product. [tex](a.b)^{c}=a^{c}.b^{c}[/tex]Zero power. a⁰ = 1Negative exponent. [tex]a^{-b}=\frac{1}{a^{b}}[/tex]Note that the first 3 rules use the same base. If there are more than 1 base number, then group the exponent that have the same base first.
Example:
[tex](2^{3}.3^{-2}) : (2^{-3}.3^{2})=( 2^{3}:2^{-3}).(3^{-2}:.3^{2})[/tex]
Tips: To simplify exponent numbers, sometimes it is useful to modify the base whenever possible.
Example:
8².2⁻⁵ = (2³)². 2⁻⁵ = 2⁶. 2⁻⁵ = 2⁶⁻⁵ = 2¹ = 2
The given problem:
Simplify (2⁴) (2⁵)
Use the product rule:
2⁴ . 2⁵ = 2⁴⁺⁵ = 2⁹
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Use graphing to find the x-coordinate of the solution to each system. ) x + 2y = 6 5x - 4y = 16 A) -1 B) 4 C) 5 D) 1
The x coordinate of the solution to the system is 4
What are linear equations?Linear equations are equations that have constant average rates of change. Note that the constant average rates of change can also be regarded as the slope or the gradient
How to determine the solution to the system?A system of linear equations is a collection of at least two linear equations.
In this case, the system of equations is given as
x + 2y = 6
5x - 4y = 16
Next, we plot the graph of x + 2y = 6 and 5x - 4y = 16 (see attachment)
From the attached graph, the point of intersection is
(x, y) = (4, 1)
Remove the y coordinate
x = 4
Hence. the x coordinate of the solution to the system is 4
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When finding the intersection of two lines from both Algebra I and Geometry, you first "set the linear equations equal to each other. Find the intersection Doint of the two lines whose equations are shown below. Be sure to find both the x and y coordinates. y=5x+1 and y=2x -11
The intersection point of two lines are (-4 , -19)
The intersection point are those point where two line met of cut each other are called intersection point.
the given line of equation are
y = 5x + 1
y = 2x - 11
On solving these two equation we get the values as follow,
value of x = -4
value of y = -19
hence by this we can say the the intersection point of the above lines are
( -4 , -19 )
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-1/5 - (1/2 + -1/4) divided by 1/3
The answer of -1/5 - (1/2 + -1/4) divided by 1/3 is [tex]-\frac{27}{20}[/tex]
A formula is a statement containing at least two numbers/variables and at least one mathematical operation. An expression is a combination of terms combined by mathematical operations such as subtraction, addition, multiplication, and division.The terms that appear in the formula are:- Constant: A constant is a fixed number. Variables: Variables are symbols that do not have fixed values. Term: A term can be a single constant, a single variable, or a combination of variables and constants combined with multiplication or division. Coefficient: A coefficient is a number that is multiplied by a variable in an expression.We have given an expression [tex]-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))[/tex] .
On simplifying , we get
[tex]-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))=-\frac{1}{5}-(\frac{1}{2} -\frac{1}{4} )\\\\-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))= - \frac{1}{5}-(\frac{2-1}{4} )\\\\\-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))= - \frac{1}{5}-\frac{1}{4} \\\\-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))=\frac{-4-5}{20} \\\\-\frac{1}{5}-(\frac{1}{2} +(-\frac{1}{4} ))=\frac{-9}{20}[/tex]
Now , we are asked to divide -9 /20 by 1 /3 .
On dividing, we get
[tex]\frac{\frac{-9}{20} }{\frac{1}{3} } =\frac{-9}{20} \times\frac{3}{1} \\\\=\frac{-27}{20}[/tex]
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A school trip uses a 42-seater coach (excluding the driver’s seat).
36 students, 3 teachers and 3 parents go on the coach.
Each student pays £4.25 towards the cost of the coach. Each parent pays twice as much
as a student pays. The teachers travel for free.
The coach company charges £195 for the trip.
a Show that the money paid by the students and parents is not enough to pay for the
coach.
b How much should each teacher pay so that the cost of the coach is covered exactly?
c On the day of the trip, 8 students are ill and they get their money back. The
passengers on the coach share the extra cost equally. What is the extra payment for
each passenger?
Step-by-step explanation:
the total charge is £195.
we have 36 students. each paying £4.25.
we have 3 parents. each paying 2×£4.25 = £8.50
we have 3 teachers. each paying £0.
a.
the payments are
36×4.25 + 3×8.50 = 153 + 25.5 = £178.50
that is clearly below the total charge of £195.
there is a deficit of 195 - 178.5 = £16.50
b.
3 teachers.
so, each should pay
16.5 / 3 = £5.50
c.
8 students drop out.
that means
8× 4.25 = £34.00 are not paid.
we have now
28 students
3 parents
3 teachers
on the bus.
these are 34 persons.
34 / 34 = £1.00
they each would have to pay £1.00 as extra payment.
If a rectangle has sides
5x + 7 and 2x + 2. What is the perimeter of the rectangle in terms of x in simplest
form?
Answer:
14x+18
Step-by-step explanation:
2(5x+7) + 2(2x+2)
10x+14+4x+4
14x+18
Tell which set or sets the 8 belongs to: natural numbers, whole, numbers, integers, rational numbers, irrational numbers, or real
numbers.
The number 8 is a natural number, whole number, integer, rational number, real number.
All positive numbers from 1 to infinity are considered natural numbers and are therefore a component of the number system. Because they don't contain zero or negative numbers, natural numbers are also known as counting numbers. They are a subset of real numbers, which only include positive integers and exclude negative, zero, fractional, and decimal numbers.
All natural numbers and 0 are included in the category of whole numbers. They are a subset of real numbers, which exclude negative numbers, decimals, and fractions. Whole numbers include counting numerals as well. In this post, let's study all there is to know about whole numbers.
The collection of positive and negative integers is known as an integer. Integers, like whole numbers, do not include the fractional portion.
Irrational number, any real number that cannot be expressed as the quotient of two integers—that is, p/q, where p and q are both integers.
Real number is a value of a continuous quantity that can represent a distance along a line. Real numbers can be defined as the union of both rational and irrational numbers.
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Solve each equation. Check for extraneous solutions. 5(3 x+1)¹/₄=10
The extraneous solution for the equation 5(3 x + 1)^ ( 1 / 4) = 10 is x = 5.
We are given an equation:
5(3 x + 1)^ ( 1 / 4) = 10
Divide each side by 5, we get that:
⁴√3 x + 1 = 2
Raise both sides to the power 4, we get that:
3 x + 1 = 2⁴
3 x + 1 = 16
3 x = 16 - 1
3 x = 15
x = 15 / 3
x = 5
Therefore, we get that, the extraneous solution for the equation 5(3 x + 1)^ ( 1 / 4) = 10 is x = 5.
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Mario walks 9 blocks from his home to a resturant. He then walks back toward home for 5 blocks, where he stops at a bookstore. How many blocks is Mario from his home?
Answer:
4
Step-by-step explanation:
9 - 5 = 4
This figure is a step in which construction?
a. What is the solution of √5 x-1 +3=x ? Check your results.
The solution is 10
Firstly rewriting the equation to frame the steps of solution -
√(5x-1) + 3 = x
Shifting 3 to the Right Hand Side of the equation -
√(5x-1) = x - 3
Now taking square of both sides of the equation
(√(5x-1))² = (x - 3)²
On solving the square we get the equation -
(5x - 1) = x² + 9 - 6x
Further solving the equation -
5x - 1 - 9 = x² - 6x
5x - 10 + 6x = x²
x² - 11x + 10 = 0
Solving the equation by factorizing -
x² - 1x - 10x + 10 = 0
x (x - 1) - 10(x - 1) = 0
(x - 1) (x - 10) = 0
x = 1, 10
Keeping the value of x in equation to find the solution and check the results-
x = 1
√(5×1-1) +3 = 1
√(5 - 1) + 3 = 1
√4 + 3 = 1
2 + 3 = 1
5 = 1
x = 10
√(5×10-1) +3 = 10
√(50 - 1) + 3 = 10
√49 + 3 = 10
7 + 3 = 10
10 = 10
Since, 10 = 10, the result is correct.
Hence, the solution is 10
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For each annual rate of change, find the corresponding growth or decay factor. -75 %
For each annual rate of change , the corresponding growth or decay factor is - 0.75 .
Given : annual rate of change = - 75 % .
To find : corresponding growth or decay factor .
For an exponential growth function ,
y = a ( [tex]1 + r^{t}[/tex] ) ,
( 1 + r ) is the growth factor and
for an exponential decay function y = a ( [tex]1 - r ^{t}[/tex] ) ,
( 1 - r ) is the decay factor .
According to question ,
( - 75 ) / 100 = - 0 .75
Hence , for each annual rate of change , the corresponding growth or decay factor is - 0.75 .
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Jackson has a points card for a movie theater. • He receives 35 rewards points just for signing up. • He earns 8.5 points for each visit to the movie theater. • He needs 154 points for a free movie ticket. Write and solve an equation which can be used to determine x, the number of visits Jackson must make to earn a free movie ticket.
The number of visits Jackson must make to earn a free movie ticket is 14 and the equation is 8.5x + 35 = 154.
What is a linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
It is given that:
Jackson has a points card for a movie theater. He receives 35 reward points just for signing up. He earns 8.5 points for each visit to the movie theater.
Let x be the number of visits Jackson must make to earn a free movie ticket.
8.5x + 35 = 154
8.5x = 154 - 35
8.5x = 119
x = 119/8.5
x = 14
Thus, the number of visits Jackson must make to earn a free movie ticket is 14 and the equation is 8.5x + 35 = 154.
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The population of a community, , is modeled by this exponential function, where x represents the number of years since the population started being recorded. what is the approximate population 3 years after the population started being recorded?
The approximate population 3 years after the population started being recorded is 2584.
What is an exponential function:
An exponential function is a Mathematical function in the form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.
Given that,
The population of a community is modeled by this exponential function, where x represents the number of years since the population started being recorded:
p(x) = 2,400(1.025)×
Plug x = 3 years in the above function.
P(3) = 2400(1.025)³
P(3) = 2400x1.0768
P(3) = 2584.53 ≈ 2584 people
Therefore,
The approximate population 3 years after the population started being recorded is 2584 people.
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What is each logarithm expanded?
a. log₃ 250/37
We need to know about logarithm to solve this problem. The logarithm expanded is [tex]\frac{log 250}{log 3} -\frac{log 37}{log 3}[/tex]
Logarithm is the power to which a base should be raised to get a given number. Expanding logarithm is writing logarithms usisng addition, subtraction, simplifying the logarithm. In logarithm, log ab can be written as log a+ log b and log a/b can be written as log a -log b, [tex]log_{a} b[/tex] can be written as log b/log a. We will be using these rules to expand the given logarithm. In the given logarithm we have the base of the log as 3 and we have logarithm of fraction, which we can expand.
[tex]log_{3} \frac{250}{37} =log_{3} 250 - log_{3} 37=\frac{log 250}{log 3} -\frac{log 37}{log 3}[/tex]
So we found the expansion of the given algorithm using the rules for expansion of algorithm to be [tex]\frac{log 250}{log 3} -\frac{log 37}{log 3}[/tex].
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put these numbers in least to greatest form7.25, -9.8, 0
Answer:
7.25, 0, -9.8
Step-by-step explanation:
For each pair of functions, find f(g(x)) and g(f(x)) .
f(x)=3 x²+2, g(x)=2 x
The values for
[tex]f(g(x)) = 6x^3 +2 \\g(f(x)) = 6x^2 + 4[/tex]
What are the domain of the function f(x)=3 x²+2, g(x)=2 x ?[tex]Given : f(x)=3 x²+2, g(x)=2 x[/tex]
1.
[tex](f.g)(x) = f(g(x)) \\g(x) = 2x \\f(x) = 3 x^2+2 \\f(2x) = 6x^3 +2[/tex]
2.
[tex](g.f)x = g(f(x)) \\fx = 3x^2 + 2 \\g(3x^2 + 2) = 2 (3x^2 + 2) = 6x^2 + 4[/tex]
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Please help please I'll give brainlyest
[tex]\bf \implies{The \: correct \: option \: is \: (c) \: 240}[/tex]
Step-by-step explanation:[tex] \large \frac{40}{5} . \: 6 \: . \: \frac{10}{2} [/tex]
Step 1) : Cross out the common factor
[tex] \bf \longrightarrow\frac{ \cancel{40}}{ \cancel{5}} . \: 6 \: . \: \frac{ \cancel{10}}{ \cancel{2} } \\ \\ \bf \longrightarrow8 \times 6 \times 5[/tex]
Step 2 : Multiply it
[tex] \bf \longrightarrow8 \times 6 \times 5 \\ \bf \longrightarrow8 \times 5 \times 6 \\\bf \longrightarrow 40 \times 6 \\ \bf \longrightarrow240[/tex]
what relationship must hold for the point to be equidistant from the origin and the xz-plane? make sure that the relationship you state is valid for positive and negative values of a, b, and c
The relationship must hold for the point to be equidistant from the origin and the x z-plane is a^2+b^2+c^2 > b^2.
What is equidistant point:
A point is said to be equidistant from two other points when it is at an equal distance away from both of them. For example, the perpendicular bisector of a line segment is equidistant from both endpoints.
Now,
distance from P(a, b, c) and the Origin (0,0,0)
[tex]\sqrt{(a-0)^2+(b-0)^2+(c-0)^2} =\sqrt{a^2+b^2+c^2}[/tex]
distance from p(a ,b, c) and the x-z plane
[tex]\sqrt{(0-0)^2+(b-0)^2+(0-0)^2} = |b|[/tex]
Here,
P is equidistant from the Origin and the x - z plane
[tex]\sqrt{a^2+b^2+c^2} = |b|[/tex]
[tex]a^{2} +b^2+c^3 = b^2[/tex]
[tex]a^{2} +c^2 = 0[/tex]
Hence a=c=0 ,thus the point P must be of the form (0,b,0) for any b ∈R
Again,
point P far the from the origin than from the x z plane
[tex]\sqrt{a^2+b^2+c^2} > |b|[/tex]
[tex]a^{2} +b^2+c^2 > b^2[/tex]
[tex]a^{2} +c^2 > 0[/tex]
Hence a, c ∈R such that atleast one of the a ,b ,c is non-zero.
Therefore,
(a ,b ,c) for any b∈ R and for a, c ∈ R with atleast one of the a and c is non-zero.
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What type of function is this based off of these data points:
(-6,-726)
(-2,-34)
(0,0)
(2,18)
(6 ,582)
Based on the data points given, the type of function shown is a Quadratic function.
What is a quadratic function?A quadratic function is one where the values of x and y form data points that lead to a parabola being formed.
This parabola forms when the values of the data points start away from 0 and then reach zero and rise again. This is what happens with these points so the function must be a quadratic function.
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2. The population of a city grows at a rate of 5% per year. The population in 1990 was
400,000. In what year would we predict the population to reach 1,000,000?
The year in which we predict the population to reach 1,000,000 based on annual growth rate of 5% is 2009
How many years would population will be 1,000,000?
The year that the population of the city would reach 1,000,000 can be determined by first of all computing the number of years beginning from 1990 that it would take the population size of 400,000 to reach 1,000,000
FV=PV*(1+G)^n
FV=future population size=1,000,000
PV=present population size=400,000
g=population growth rate per year=5%
N=number of years it takes population to become 1,000,000=unknown
1,000,000=400,000*(1+5%)^N
1,000,000/400,000=(1+5%)^N
2.50=(1.05)^N
take log of both sides
ln(2.50)=N*ln(1.05)
N=ln(2.50)/ln(1.05)
N=19(rounded to the nearest whole number)
number of years before population reach 1m=1990+19
number of years before population reach 1m=2009
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A quadratic function is shown on the graph.
an upward opening parabola beginning with closed circle at negative 2 comma 4, which decreases to a vertex at 0 comma 0 and then increases to an open circle at 3 comma 9
What is the range of the function?
{x | −2 ≤ x < 3}
{x | −2 < x ≤ 3}
{y | 0 ≤ y < 9}
{y | 0 < y ≤ 9}
The range of the quadratic function is {y |0 ≤ y < 9}
What is the range of a function?The range of a function is the set of output values of the graph. This in other words mean the range of a function is the set of y values of the graph.
How to determine the range of the function?From the question, we have the following parameters:
The parabola opens upwardVertex = (0, 0)Closed circle point = (-2, 4)Open circle point = (3, 9)Write out the vertex
So, we have
Vertex = (0, 0)
Remove the x value
y = 0
From the question, we have
The parabola opens upward
This means that the vertex is a minimum
So, the range is
y ≥ 0
Rewrite as
0 ≤ y
Recall that:
Closed circle point = (-2, 4)
Open circle point = (3, 9)
Remove the x values
Closed circle point = 4
Open circle point = 9
Remove the smaller y value
Open circle point = 9
This means that the highest value of y is at open circle point = 9
This is represented as
y < 9
So, we have
0 ≤ y and y < 9
Combine both
0 ≤ y < 9
Rewrite properly as
{y |0 ≤ y < 9}
Hence, the range of the quadratic function is {y |0 ≤ y < 9}
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{y |0 ≤ y < 9}
The person above need the shine, im just making it simple
first to answer will get brainliest
Reflexive Property of congruence justifies the statement
∠K ≅ ∠K
The Reflexive Property of congruence states that any geometric figure is congruent to itself. The figures are being the reflection of itself. They have the line segments of same length, an angle has the same angle measure and the figure has same shape and size as itself.
If two triangles have a common side or a common angle, then Reflexive Property of congruence is used to prove that the two triangles are congruent.
Since the given angle is K, therefore by the Reflexive Property of congruence we could say that the angle K is congruence to angle K.
⇒ ∠K ≅ ∠K
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Charlotte is visiting Brazil and is looking at souvenirs to purchase. She finds a jersey that costs 110 Brazilian real. If the current exchange rate is 1 dollar:5.43 Brazilian real, how much does the jersey cost in U.S. dollars?
a. $4.94
b. $20.26
c. $104.57
d. $597.30
The points H, I, J and K all lie on the same line segment, in that order, such that the ratio of HI: IJ: JK is equal to 1:2:2. If HK=45, find HJ.
Will give brainliest
When the points H, I, J and K all lie on the same line segment, in that order, such that the ratio of HI: IJ: JK is equal to 1:2:2 ,the value of HJ is 27.
How to illustrate the information?Fronxrye information, the points H, I, J and K all lie on the same line segment, in that order, such that the ratio of HI: IJ: JK is equal to 1:2:2.
Therefore, the value of HJ will be:
= (1 + 2)/(1 + 2 + 3) × 45
= 3/5 × 45
= 27
Therefore, the value of HJ is 27.
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-can someone help me 4d+3 if d=-1
Answer:
7
Step-by-step explanation:
-4d+3
-4(-1) + 3 Insert -1 for d
4+3=7
Determine if 0.131331333133331333331... is rational or irrational and give a reason for your answer.
A road race is 13 1 2 miles long. there are water stations every 3 4 mile including at the finish line. how many water stations are there in all?
By solving the improper fractions, there are 18 water stations in all.
This problem can be solved by two methods:
Method 1 :
First of all, calculate 13²/₃ miles in fraction. The value is 13.5 miles. Then calculate 3/4 mile in fraction. The value is .75 miles.
Now divide 13.5 miles by 0.75 miles.
13.5 ÷ 0.75 = 18
So, 18 water stations are there in all.
Method 2 :
Convert 13 ¹/₂ miles into an improper fraction from 27/2 miles.
Now, again divide 27/2 miles by 3/4.
27/2 ÷ 3/4 = ²⁷/₂ ₓ ⁴/₃ = 18
So, 18 water stations are there in all.
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What is the volume of a cone (in cubic inches) of radius 2 inches and height 6 inches? Use 3.14 for π. Round your answer to the nearest hundredth. (1 point) a 12.56 b 25.12 c 37.68 d 75.3
Given the radius and height, the volume of the cone to the nearest hundredth is 25.12 in³.
Option b)25.12 in³ is the correct answer.
What is the volume of the cone with the given radius and height?A cone is simply a 3-dimensional geometric shape with a flat base and a curved surface pointed towards the top.
The volume of a cone is expressed as;
V = (1/3)πr²h
Given the data in the question;
Radius r = 2inHeight h = 6inConstant pi π = 3.14Volume V = ?Plug the given values into the volume formula above and simplify.
V = (1/3)πr²h
V = (1/3) × 3.14 × (2in)² × 6in
V = (1/3) × 3.14 × 2in² × 6in
V = (1/3) × 3.14 × 4in² × 6in
V = 25.12 in³
Given the radius and height, the volume of the cone to the nearest hundredth is 25.12 in³.
Option b)25.12 in³ is the correct answer.
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Which angle number represents the linear pair of \angle FHC∠FHC?
The angle number that represents the linear pair of angle FHC is: angle 5.
What is the Linear Pair?The linear pair consists of two adjacent angles that are on a straight line, whose sum will be equal to 180 degrees when added together.
Thus, the two adjacent angles share a common side and a common vertex, and they are supplements of each other. That is, they have a sum of 180 degrees.
Angle 4 represents angle FHC. Angle 5 is adjacent to angle 4, and both lie on a straight line.
Therefore, angle FHC + angle 5 = 180 degrees.
In conclusion, we can state that, the angle number that represents the linear pair of angle FHC is: angle 5.
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For questions 4-7, find the measures of the angles in the picture below.
(Questions shown in image)
Answer:
Measures of each angle are as follows:
4. ∠BXC = 20°
5. ∠FXE = 85°
6. ∠GXD = 75°
7. ∠BXF = 75°
Step-by-step explanation:
#4
∠BXC is opposite ∠EXD and so it is the same as ∠EXD = 20°
#5
∠FXE is opposite ∠CXG and so it is the same as ∠CXG = 85°
#6
Angles ∠CXG, ∠GXD and ∠EXD line on a straight line so the sum of these angles must be 180°
∴ m∠CXG + m∠GXD + m∠EXD = 180°
Substituting known values for ∠CXG and ∠EXD we get
85 + ∠GXD + 20 = 180°
105 + m∠GXD = 180°
m∠GXD = 180-105 = 75°
#7
∠BXF is opposite ∠GXD which we computed in #6 to be 75°
So m∠BXF = m∠GXD = 75°