Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. (x+4)⁸ is expanded as x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536
What is meant by binomial expansion theorem?The Binomial theorem tells us how to expand expressions of the form (a + b)ⁿ, for example, (x + y)⁷. The larger the power is, the harder it is to expand expressions like this directly.
The total number of terms in the expansion of (x+y)ⁿ are (n+1).
The sum of exponents of x and y is always n.
nC₀, nC₁, nC₂,….., nCₙ are called binomial coefficients and also represented by C₀, C₁, C₂, ….., Cₙ
The binomial coefficients which are equidistant from the beginning and from the ending are equal.
i.e. nC₀ = nCₙ, nC₁ = nCₙ₋₁ , nC₂ = nCₙ₋₂ ,….. etc.
Let the expression be (x + 4)⁸
Use the binomial expansion theorem to find each term. The binomial theorem states
(a+b)n=n∑k=0nCk⋅(an−kbk)(a+b)n=∑k=0nnCk⋅(an-kbk).
8∑k=08!(8−k)!k!⋅(x)8−k⋅(4)k∑k=088!(8-k)!k!⋅(x)8-k⋅(4)k
Expand the summation
8!/(8−0)!0!.(x)⁸⁻⁰⋅(4)⁰+8!/(8−1)!1!(x)⁸⁻¹⋅(4)¹+…+8!/(8−8)!8!(x)⁸⁻⁸⋅(4)⁸
Simplify the exponents for each term of the expansion.
1⋅(x)⁸⋅(4)⁰+8⋅(x)⁷⋅(4)1+…+1⋅(x)⁰⋅(4)⁸¹⋅(x)⁸⋅(4)⁰+8⋅(x)⁷⋅(4)¹+…+1⋅(x)⁰⋅(4)⁸
Simplify the polynomial equation, we get
x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536
Therefore, by expanding the binomial equation, we get
(x + 4)⁸ = x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536
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Expand each logarithm.
log₃ 7(2 x-3)²
If the expression is log₃ 7(2 x-3)², then the expansion of the expression is [[tex]log_{3}[/tex] 7+ [tex]log_{3}[/tex] [tex](2x-3)^{2}[/tex]]
Given that the expression is log₃ 7(2 x-3)².
We are told to find the expansion of the expression.
Logarithmic function is basically the opposite of the exponential function and is denoted by log or ln.
It says that [tex]log_{a} b[/tex]= [tex]log_{x}b /log_{x}a[/tex].
The expression is log₃ 7(2 x-3)².
We know that log mn=log m +log n.
log₃ 7(2 x-3)²=[[tex]log_{3}[/tex] 7+ [tex]log_{3}[/tex] [tex](2x-3)^{2}[/tex]]
We know that log [tex]a^{b}[/tex]=log b/log a but we are not changing here,So,
log₃ 7(2 x-3)²=[[tex]log_{3}[/tex] 7+ [tex]log_{3}[/tex] [tex](2x-3)^{2}[/tex]]
Hence if the expression is log₃ 7(2 x-3)², then the expansion of the expression is [[tex]log_{3}[/tex] 7+ [tex]log_{3}[/tex] [tex](2x-3)^{2}[/tex]]
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A line passes through the points (6, -3) and (j, -4). If the slope of the line is 1/3 find the value of j.
Answer:
j = 3
Step-by-step explanation:
Recall the slope formula:
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Now, simply substitute the values from the question.
[tex]\frac{1}{3} =\frac{-4+3}{j-6}[/tex]
Now, use the cross-multiplication method. (remember, when subtracting negatives, the negative becomes a positive)
[tex]1(j-6)=3(-4+3)[/tex] Cross Multiplication
[tex]1(j-6)=3(-1)[/tex] Simplify
[tex]j-6=-3[/tex] Distributive Property
[tex]j=3[/tex] Addition Property
Therefore, j = 3.
Write each equation in logarithmic form.
1/4=4⁻¹
The logarithmic form of 1/4 = 4⁻¹ is log₄ 4⁻¹ = ₋1.
The fundamental logarithmic function has the notation f(x) = log(r) y = log(x), where a > 0. The exponential function ay = x's inverse is represented by this. The common logarithm (log) or natural logarithm (ln) are examples of log functions (log).
The exponential function of the type aˣ = N can be converted into the logarithmic function logₐN = x. A Napier logarithm table can be used to determine the logarithmic value of any number. Logarithms are often calculated using a base of 10, therefore it is possible to get the logarithmic value of any number.
we have the exponential form as aˣ = N
1/4 = 4⁻¹
now the logarithmic form is:
logₐ N = x
log₄ 4⁻¹ = ₋1
hence we get the logarithmic form as log₄ 4⁻¹ = ₋1.
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How do you answer and write this out correctly
2.
x^2-4y
/2
4^2- (4*-3)
/2
16+12
/2
28/2
14
During a sale, a store offered a 20% discount on a camera that originally sold for $370. After the sale, the discounted price of the camera was marked up by 20%. What was the price of the camera after the markup? Round to the nearest cent.
Answer:
624
Step-by-step explanation:
650 * .80 = 520520 * 1.20 = 624
f(x)=√x-4
y = √x=4
How to find the inverse function
f ⁻¹( x)= x^2+4 is the inverse of f(x)= [tex]\sqrt{x-4}[/tex]
what is an inverse function ?
An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by 'f' or 'F', then the inverse function is denoted by f-1 or F-1.
write f(x)= [tex]\sqrt{x-4}[/tex] as an equation
y= [tex]\sqrt{x-4}[/tex]
interchange the variables
x= [tex]\sqrt{y-4}[/tex]
solve the equation for y we get
squaring both side ,we have
x^2 = y-4
this implies that
y= x^2 +4
replace y with f ⁻¹( x) to show the final answer
f ⁻¹( x)= x^2+4
Hence, f ⁻¹( x)= x^2+4 is the inverse of f(x)= [tex]\sqrt{x-4}[/tex]
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What expression is equivalent to -3(2x-4)
The given expression , the value of x is 2.
given,
equivalent expression = -3(2x - 4)
= -6x + 12
x = 2
Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.
3(x + 2) and 3x + 6 are equivalent expressions because the value of both the expressions remains the same for any value of x. 3x + 6 = 3 × 4 + 6 = 18. and can also be written as 6(x2 + 2y + 1) = 6x2 + 12y + 6.you can write equivalent expressions by combining like terms. Like terms are terms that have the same variables raised to the same powers. For example, the list shows some pairs of like terms. the new coefficient is 9.Two expressions are equivalent if they can be simplified to the same third expression or if one of the expressions can be written like the other. In addition, you can also determine if two expressions are equivalent when values are substituted in for the variable and both arrive at the same answer.to know more about equivalent expression;
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2y-10x=14 in slope intercept form
Answer:
You want to start off my isolating the y onto one side:
2y-10x=14 ---> 2y=14+10x
then you remove the coefficient from the left side:
2y=14+10x -----> y=7+5x
Then all you want to do is rearrange the equation to match slope intercept:
y=7+5x ----> y=5x+7
Answer:
You want to start off my isolating the y onto one side:
2y-10x=14 ---> 2y=14+10x
then you remove the coefficient from the left side:
2y=14+10x -----> y=7+5x
Then all you want to do is rearrange the equation to match slope intercept:
y=7+5x ----> y=5x+7
Step-by-step explanation:
Simplify each expression.
ln e¹⁰
We will get lne¹⁰ itself after simplifying the given expression lne¹⁰.
What is an expression?In mathematics, an expression is a sentence that contains at least two numbers or variables and at least one math operation. Addition, subtraction, multiplication, or division are all examples of math operations. An expression's structure is as follows: (Number/variable, Math Operator, Number/variable) is an expression.So,
Given expression: ln e¹⁰Then,
lne¹⁰⇒ lne¹⁰Therefore, we will get lne¹⁰ itself after simplifying the given expression lne¹⁰.
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Simplify. √75 + √125
The value of the expression √75 + √125 is 5 (√3 + √5 ).
A number is a fundamental mathematical unit. For counting, measuring, and comparing quantities, we utilize numbers. A set of symbols, or numerals, used to represent numbers is known as a number system.
A square root is a factor of a number that yields the original number when multiplied by itself.
Consider the expression √75 + √125,
Let's assume x = √75 + √125
Now,
75 = 5 × 15 = 5 × 3 × 5
125 = 5 × 25 = 5 × 5 × 5
Therefore,
x = √75 + √125
x = √ [ 5 × 3 × 5] + √ [ 5 × 5 × 5 ]
x = 5√3 + 5√5
x = 5 × [ √3 + √5 ]
x = 5 × [ 1.73 + 2.24 ]
x = 5 × 3.97
x = 19.85
√75 + √125 = 19.85
Hence, the value of √75 + √125 is equal to 19.85.
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| | x+4|-7 |< 5, | | = absolute value
The absolute value of the inequality is x < 6
What is absolute value?Absolute value can be defined as the describing the distance of a number from zero on the number line, without considering direction.
It is also the value of the number with no consideration to it's sign.
Given the inequality;
| | x+4|-7 |< 5 | |
Take the absolute value and expand the bracket
(x + 4) + 7 < 5
expand the bracket
x + 4 + 7 < 5
x + 11 < 5
Make 'x' the subject
x < 5 - 11
x < -6
Take the absolute value
x < 6
Thus, the absolute value of the inequality is x < 6
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Pre-calculus Help please
The inverse of the expression y = 5 · (x - 6)² - 5 is x = √[(y + 5) / 5] + 6. (Correct choice: D)
How to determine the inverse of a function
According to the statement we have a function of the form y = f(x) and we need to find its inverse, that is, finding the equation of x in terms of y. This can be done by using algebra properties:
y = 5 · (x - 6)² - 5 Given
y = 5 · [(x - 6)² - 1] Distributive property
y / 5 + 1 = (x - 6)² Compatibilities with multiplication and addition / Existence of additive and multiplicative inverses / Associative and modulative properties
x - 6 = √(y / 5 + 1) Definition of square root / Symmetric property
x = √(y / 5 + 1) + 6 Compatibility with addition / Existence of additive inverse / Associative, commutative and modulative properties
x = √[(y + 5) / 5] + 6 Addition of fractions with different denominators / Result
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Unit cube:
What is the volume of the following figure?
cubic units
Front View
Back View
Answer:
back view
Step-by-step explanation:
You are very very very correct
PLease help!!!! 3( - 4y - 5) = 8y + 25, Show your work please!!
Answer:
y = -2
Step-by-step explanation:
Given equation:
[tex]3(-4y-5)=8y+25[/tex]
Distribute:
[tex]\implies 3 \cdot -4y - 3 \cdot 5=8y+25[/tex]
[tex]\implies -12y - 15=8y+25[/tex]
Add 15 to both sides:
[tex]\implies -12y - 15+15=8y+25+15[/tex]
[tex]\implies -12y =8y+40[/tex]
Subtract 8y from both sides:
[tex]\implies -12y-8y =8y+40-8y[/tex]
[tex]\implies -20y =40[/tex]
Divide both sides by -20:
[tex]\implies \dfrac{-20y}{-20} =\dfrac{40}{-20}[/tex]
[tex]\implies y =-2[/tex]
Now we have to,
→ solve for the required value of y.
Let's solve the problem,
→ 3(-4y - 5) = 8y + 25
→ -12y - 15 = 8y + 25
→ -12y - 8y = 25 + 15
→ -20y = 40
→ y = -40/20
→ [ y = -2 ]
Hence, the value of y is -2.
Factor each expression.
5 x²+13 x-6
The answer to the present question is:5x² + 13x - 6 = (5x - 2) (x + 3)
For a polynomial, ax² + bx + c, we want to separate the center term because the sum of these two numbers, whose sum is adequate to the merchandise of the opposite two numbers.
So, here, we'd like to rewrite 13 because the sum of two numbers, whose product is adequate to the opposite two numbers, i.e., 5*(-6) = -30
So, to separate 13, two numbers used here will be: 1 5 and -2, because after we will add 15 and -2 we are going to get 13 and that we will multiply them, then we are going to get
-30.5x² + 13x -6 = 5x² + 15x - 2x - 6
= 5x(x + 3) - 2(x + 3)
= (5x - 2) (x + 3)
Therefore, 5x² + 13x - 6 = (5x - 2) (x + 3)
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Help ASAPPP explain why Answer is correct
Answer:
Option 2
Step-by-step explanation:
The distances will be the same because the bisector will create congruent triangles, and the desired equalities follow from CPCTC and the definition of congruency.
The distance between the school and Target is 4.2 miles. On a map the ratio is 0.5 inches is equal to 3 miles. How far would the school and Nokesville be on the map?
The the distance between the school and Nokesville be on the map is 0.7 inches.
What is scaling ?
It is the technique to resize drawing with comparing to actual size of object which uses proportion or ratio. For example, Mapping of earth, drawing prototype of automobiles.
It is given that :
The distance between the school and Target is 4.2 miles and on a map the ratio is 0.5 inches is equal to 3 miles that is :
3 miles = 0.5 inches on map
Now to find the distance between the school and Nokesville be on the map we first find for 1 miles on the map and then multiply it by 4.2 miles :
3 miles = 0.5 inches on map
1 miles = 0.5/3 inches on map
4.2 miles = (0.5/3) x 4.2 inches on map
4.2 miles = 0.7 inches inches on map
Therefore, the the distance between the school and Nokesville be on the map is 0.7 inches.
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Please help is this true or false
the answer for your question is true. I believe.
Factories (a + b)³-3a²b(a + b)
Answer:
(a+b) (a+2ab+b2−3a2b)
Step-by-step explanation:
Will the answer to the addition problem below be odd or even?
742 +61 = ?
A. Odd
B. Even
Answer:
odd
Step-by-step explanation:
Not sure tho
Write the following expression in exponential form: 8x8x8x8x3x3
Answer:
It is: 8^4 times 3^2 = 36,864
Answer:
8^4 x 3^2
Step-by-step explanation:
There's 4 8's and 2 3's so that's the number the exponents will be
X = 2y + 18
5x = -3y - 27
Use the substitution method to solve the system below
Answer:
(0, - 9 )
Step-by-step explanation:
x = 2y + 18 → (1)
5x = - 3y - 27 → (2)
substitute x = 2y + 18 into (2)
5(2y + 18) = - 3y - 27
10y + 90 = - 3y - 27 ( add 3y to both sides )
13y + 90 = - 27 ( subtract 90 from both sides )
13y = - 117 ( divide both sides by 13 )
y = - 9
substitute y = - 9 into (1)
x = 2(- 9) + 18 = - 18 + 18 = 0
solution is (0, - 9 )
-15 divided by 0.3 and also -4 divided by -20
Answer:
first one = -50
second one = 0.2
Step-by-step explanation:
Answer: bro if your on a computer or phone just use a calculator lol but ill put the answer below
Step-by-step explanation:
-15 divided by 0.3= -50
-4 divided by -20 = -24
f(x) = 3x²5x + 20
Find f(-8)
Answer:
f(-8)= -7660
Step-by-step explanation:
f(-8)= 3(-8)^2·5(-8)+20
Please answer the question in the photo! ( brainliest )
Answer:
Step-by-step explanation:
basically the angle is a 90 degree angle and it tilted slightly to the left measures that you come out with 3/4 .
what is the vertex of this graph
(minimum or maximum)
Answer:
maximum at (0, 2 )
Step-by-step explanation:
the vertex is the turning point on the graph.
the graph increases on the left up to (0, 2 ) then decreases
this indicates the vertex is a maximum
If an airplane can travel at 20,000 miles per hour, how many meters per second does it travel? Knowing that: 1 mi 1.609 km
Answer: 8,938 [tex]\frac{8}{9}[/tex] or 8,938.8889 meters per second
Step-by-step explanation:
To solve this problem, we will use a process called dimensional analysis (that you have probably done many times without realizing it).
What is dimensional analysis? Let's say you know there are 12 inches in a foot, and you want to know how many inches are in 2 feet. 12 * 2 = 24 inches in a foot, that's dimensional analysis!
Here is the process you would take to solve your problem:
➜ If we have the units [tex]\frac{hour}{hour}[/tex], they "cancel" out to become one. This way, we make sure we end up with the correct units, [tex]\frac{meters}{second}[/tex].
➜ We take units we know, such as 3,600 seconds in an hour, and multiply them by the original value until we end up with the measurement we want.
[tex]\displaystyle 20,000\; \frac{miles}{hour}*\frac{1 \;hour}{3,600\; seconds} *\frac{1,609\;meters}{1\;mile}=\frac{20,000*1*1,609\;meters}{3,600\;seconds} =\frac{32,180,000\;meters}{3,600\;seconds} = 8,938.89\; \frac{meters}{second}[/tex]
In other words;
8,938.89 meters per second
You and your friend decide to get an Uber ride to a concert in downtown Minneapolis to avoid the crazy traffic. Uber charges a flat rate fee of $5.50 just to hop in the vehicle and an additional $2.25 for each mile traveled. Lyft charges a flat rate fee of $3.25 just to hop in the vehicle and an additional $3 for each mile traveled. How many miles (x) does it take for the price to be the same?
Answer:
I deleted my answer bc i accidentally answered the wrong question
The average distance from Earth to the sun is about 9.3×107 miles. The average distance from Mars to the sun is about 1.4×108 miles. When both planets are at their average distance from the sun, how much farther is Mars from the sun than Earth?
A 79,000,000 mi
B 47,000,000 mi
C 107,000,000 mi
D 233,000,000 mi
f(x)=x² + 2x - 4; Find ƒ(−1)
Answer:
f(- 1) = - 5
Step-by-step explanation:
substitute x = - 1 into f(x)
f(x) = x² + 2x - 4 , then
f(- 1) = (- 1)² + 2(- 1) - 4 = 1 - 2 - 4 = - 5