Converting the radicals into exponent numbers, √3 (⁴√3) = ⁴√27
To work with radical numbers or roots, it is easier to express them in exponent:
[tex]\sqrt[n]{a^{b}}=a^{\frac{b}{n}[/tex]
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 √3 (⁴√3)
[tex]\sqrt{3}=3^{1/2}[/tex]
[tex]\sqrt[4]{3} =3^{1/4}[/tex]
Hence,
[tex]\sqrt{3}.\sqrt[4]{3} =3^{1/2}.3^{1/4}\\ =3^{1/2+1/4}=3^{3/4}\\ =\sqrt[4]{3^{3}}= \sqrt[4]{27}[/tex]
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A submarine was 200 feet below sea level. Then it dove down another 370 feet so the biologists could study a squid. After that, the submarine ascended 500 feet to get closer to the surface. What is the current depth of the submarine? Express your answer as an integer.
Answer:
70 feet
Step-by-step explanation:
Depth
200 + 370 - 500 = 70 feet
from x-2 subtract 4x +9
Answer:
I'm pretty sure it is -3x+7
A puppy weighs 2.8 pounds at birth and gained about 0.5 pounds per week. A kitten weighs 1.4 pounds at birth and gains about 1.2 pounds per week. In how many weeks will the weight of the kitten be no less than the weight of the puppy?
The weight of the kitten will be no less than the weight of the puppy in 1.4 + 1.2x ≥ 2.8 + 0.5x week.
What is Inequality?Statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions.
Given:
A puppy weighs 2.8 pounds at birth and gained about 0.5 pounds per week.
A kitten weighs 1.4 pounds at birth and gains about 1.2 pounds per week.
puppy: 2.8 + 0.5x
Kitten: 1.4 + 1.2x
As, kitten is no less than puppy.
Kitten ≥ puppy
1.4 + 1.2x ≥ 2.8 + 0.5x
hence, the weight of the kitten will be no less than the weight of the puppy in 1.4 + 1.2x ≥ 2.8 + 0.5x week.
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Tonya is paid a salary of $650 a month at her sales job. She also earns a commission on her sales in this way: 3% on all sales up to $32,000 in a month
and 8% on all higher sales. What were Tonya's total earnings for a month where her total sales were $80,000?
Tonya's total earnings for a month where her total sales were $80,000 is $5450.
What is the percentage?It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
It is given that:
Tonya is paid a salary of $650 a month at her sales job.
She also earns a commission on her sales in this way: 3% on all sales up to $32,000 in a month and 8% on all higher sales.
= 650 + 32000(3%) + (80000 - 32000)(8%)
= 650 + 960 + 3840
= $5450
Thus, Tonya's total earnings for a month where her total sales were $80,000 is $5450.
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The figure shows relationships of the dimensions of a square pool with an attached patio. Complete the equations and reasons to show that AC= BD
For the given square pool the relationship between the line segment will be BD = BC + CD and AB + BC = CD + BC and it will go to proves that AC = BD.
What is a line segment?A line section that can connect two places is referred to as a segment.
The line is here! It extends endlessly in both directions and has no beginning or conclusion.
Given that,
AB = CD
By segment addition postulate
AC = AB + BC
Again by segment addition postulate
BD = BC + CD
In addition property of equality
AB + BC = CD + BC
So,
AC = BD
Hence "BD = BC + CD and AB + BC = CD + BC will proves that AC = BD".
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a bacteria culture starts with 660 bacteria and grows at a rate proportional to its size. after 4 hours there will be 2640 bacteria. (a) express the population after t hours as a function of t. population: equation editor (function of t)
Answer:
T=x(T*2) where X=660
Step-by-step explanation:
If it has been 2 hours than t=2
plug in 2 for t and you get x*4
How much greater is 40,000 than 4,000
Answer:
40,000 is 10x greater than 4,000
Step-by-step explanation:
Times 4,000 by 10 and you get 40,000
3 (x - 17) < 9 (1 - 3x)
Answer:
x < 2
Step-by-step explanation:
3(x- 17) = 3x - 51
9(1-3x) = 9 - 27x
3x - 51 < 9 - 27x
30x - 51 < 9
30x < 60
X < 2
Show that at least three of any 25 days chosen must fall in the same month of the year.
The proof that at least three of any 25 days chosen must fall in the same month of the year holds true because; there are twelve months in a year.
What is the proof that three of any twenty-five, 25 days chosen must fall in the same month of the year as required in the task content?This proof required for the statement in the task content follows from the fact that, there are 12 months in a year.
Hence, the proof that at least three of any twenty-five, 25 days chosen must fall in the same month of the year is because, there are 12 months in a years and after choosing two days from each month, amounting to 24 days, the 25th day would fall into a month and hence, such month has 3 days already.
Ultimately, the statement that at least three of any 25 days chosen must fall in the same month of the year holds true because there are 12 months in a year.
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Sheridan Street and Richmond Street are parallel, and both are intersected by University Boulevard.
Two cars are heading west, one on Sheridan St. and one on Richmond St. Both cars turn north onto University Blvd.
What is the measure of the angle of their turn on to University Blvd.
The measure of the angle of their turn on to University Blvd is; 80°
How to interpret corresponding angles?
Corresponding angles are the angles that are formed when two parallel lines are intersected by the transversal.
Thus, from the given diagram when we apply the concept of corresponding angles, we can say that;
17x + 15 = 19x + 5
19x - 17x = 15 - 5
2x = 10
x = 5
Thus;
19x + 5 = 19(5) + 5 = 100°
Now, the sum of the angles on a straight line is 180 degrees. Thus, the measure of the angle of their turn on to University Blvd is;
180 - 100 = 80°
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Write each expression in radical form. y⁻⁴/₃
The radical expression of the term y-⁴/³ is ∛y-⁴ or ∛(1/y⁴).
What is a radical?A radical is a number that is denoted by a root and can be easily identified because before the number is the symbol √, which lets us know that it is a root.
In this case we have an exponential number since it is raised to a number, radicals can be expressed as powers raised to fractions as follows:
√a = a¹/²
If our term is raised to -4/3, it can be expressed as:
y-⁴/³ = y-¹/³ y-¹ = 1/y⁴/³ now we apply.
y⁴/³ = ∛y⁴ and apply to the numerator since it is not altered and we have
y-⁴/³ = ∛(1/y⁴)
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Micah is 7 years older than treasure. in 6 years the sum of their ages will be 51. how old is micah now?
The age of Micah is 23 years.
Let the age of treasure be x years. So, the present age of Micah will be (x + 7) years.
Now, age of treasure after 6 years = (x + 6) years
Age of Micah after 6 years = (x + 7 + 6) years
Age of Micah = (x + 13) years
As the question mentions, the sum of their ages is 51. So, relating the given information -
(x + 6) + (x + 13) = 51
Performing addition on Left Hand Side of the equation
2x + 19 = 51
Shifting 19 to Right Hand Side of the equation
2x = 51 - 19
Performing subtraction on Right Hand Side of the equation
2x = 32
Shifting 2 to Right Hand Side of the equation and performing division
x = 32 ÷ 2
x = 16 years
So, the present age of Micah = x + 7
Present age of Micah = 16 + 7
Present age of Micah = 23 years
Hence, the present age of Micah is 23 years.
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The freight elevator has the capacity of 3500 pounds if a barrel averages 50 pounds and a crate 140 pounds write an inequality that describes when x barrels and y crates will cause the elevator to be overloaded
The inequality that describes when x barrels and y crates will cause the elevator to be overloaded is 50x + 140y < 3500.
According to the given question.
Capacity of elevator = 3500 pounds
Average of a single barrel = 50 pounds
capacity of single crate = 120 pounds.
As we know that, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.
Therefore, the inequality that describes when x barrels and y crates will cause the elevator to be overloaded is given by
50x + 140y < 3500
Hence, the inequality that describes when x barrels and y crates will cause the elevator to be overloaded is 50x + 140y < 3500.
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For Comic Relief Stamford Green School are taking part in a "Bake Off".
Parents are invited to come and buy cakes. 150 parents attend: 70% donate £1 for their cake and the rest donate £2 for their cake. How much money is donated in total?
Start off with %70 of 150
0.7 • 150 = 105
To find their total donation amount,
105 • £1 = £105
Then we know the rest of the parents account for the last %30 of the attendees
We can find this out by either,
150 - 105
or
0.3 • 150
Either way we end up with 45
45 • £2 = £90
%70 (105) of the parents donated £105
%30 (45) of the parents donated £90
The parents, in total, donated £195
Use natural logarithms to solve each equation.
e x+1=30
2.4012 is the value of x after solving equation [tex]e^{x+1}[/tex] = 30 using natural logarithms
We have been asked to solve [tex]e^{x+1}[/tex] = 30 using natural logarithms
⇒ [tex]e^{x+1}[/tex] = 30
⇒ x + 1 = ㏑ 30
⇒ x = ㏑ 30 - 1
≈ 2.4012
What is a natural logarithm?A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.
Typically, the natural logarithm of x is denoted by the symbols ln x, loge x, or, if the base e is implicit, just log x. When adding parentheses for clarity, the values ln(x), loge(x), or log (x). In order to avoid ambiguity, this is notably done when the argument to the logarithm is not a single symbol.
The power to which e would need to be raised in order to equal x is the natural logarithm of x.
For instance, e2.0149... = 7.5, thus ln 7.5 equals 2.0149.
Since e1 = e, the natural logarithm of e itself, or ln e, is equal to 1, while the natural logarithm of 1 is 0, since e0 = 1.
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Correct QuestionUse natural logarithms to solve each equation.
[tex]e^{x+1}[/tex] = 30
Please help due soon.
[tex]a = \pi \: r {}^{2} \\ r {}^{2} = \frac{a}{\pi} \\ r = \sqrt{ \frac{a}{\pi} } [/tex]
Answer: First Option-7/10 + 1/5 (please help)
Answer: 2
Step-by-step explanation:
solve 4- | 2x -1 | = -3 i need it pls help
The value of x is 3
What are algebraic expressions?Algebraic expressions are expressions made up of variables, terms, factors, constants and coefficients.
They are also made up of mathematical operations like addition, subtraction, multiplication, division, etc
It is important to note that the absolute value of a number takes a positive sign.
It is denoted with the sign ' | | '
Given the expression:
4- | 2x -1 | = -3
First, take the absolute value
4 - 2x + 1 = - 3
collect like terms
2x = -3 - 4 + 1
2x = - 7 + 1
2x = -6
Make 'x' the subject of formula
x = 6/2
x = 3
Thus, the value of x is 3
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Solve the following system of equations. What is the x -coordinate of the solution?
y-3=x
y/x²-9}=1
The x-coordinate of the given equation is x=4.
What essentially is an equation?An equation is a mathematical statement composed of two representations joined by an equal sign.An example of an equation is 3x - 5 = 16.After solving this equation, we obtain the value for the variable x as x = 7.
So,
Given:
y-3 = x ...(1)y/x²-9 = 1 ...(2)Then,
y = x+3Substitute y=x+3 in equation (2):
y/x²-9 = 1 x+3/x²-9 = 1 x+3/x²-3² = 1 1/x-3=1(Cross multiply)
x-3=1x=1+3x=4Therefore, the x-coordinate of the given equation is x=4.
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Write 750 in scientific notation
Answer:
7.5 × 10[tex] {}^{2} [/tex] (scientific notation)
Step-by-step explanation:
➟ 750 = 7.5 × 10[tex] {}^{2} [/tex] (scientific notation)
For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function. f(x)=(7-x)²
Answer to this question is:
Domain of given function is (-∞,∞),{x|x∈R}
Range of given function is [0,∞),{y|y≥0}
Inverse of given function is -√x+7,√x+7
The domain is all values of x that make the expression defined.The entire range of independent variable values is the domain of a function.
This definition can be stated simply as follows: The domain is the collection of all x-values that can be used to make the function "work" and produce actual y-values.
When choosing the domain, keep in mind that a fraction's denominator (bottom) cannot be 0.
Interval Notation:(-∞,∞)
Set-Builder Notation:{x|x∈R}
The set of all valid y values constitutes the range.
Interval Notation:[0,∞)
Set-Builder Notation:{y|y≥0}
For calculation of inverse:
Interchange the variables x=(7-y)²Solve for yy=-√x+7,√x+7Replace y with f^-1(x)f^-1(x)=-√x+7,√x+7∴Domain of given function is (-∞,∞),{x|x∈R}
Range of given function is [0,∞),{y|y≥0}
Inverse of given function is -√x+7,√x+7
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SUPER URGENT!!!
Joel sells ice cream cones at the county fair. He has to rent the equipment for $36 and spend $0.52 on ingredients for each cone. What is the minimum number of ice cream cones Joel must sell at $1.40 each in order to make a profit?
Please explain ALL OF THE STEPS, or your answer is useless. Thanks!!
Denote by x the number of cones sold or produced.
Denote by C(x) the total cost.
Denote by R(x) the revenue made.
Denote by P(x) the net profit accumulated.
[tex]C_{T}(x)=C_{fixed}+C_{variable} \: \\ C_{T}(x)=36+0.52x[/tex]
[tex]R_{T}(x)=selling \: price \: x \\ R_{T}(x)=1.4x[/tex]
[tex]P(x)=R_{T}(x)-C_{T}(x) \\ P(x)=1.4x-(36 + 0.52x) \\ P(x)=1.4x - 36 - 0.52x \\ P(x)=0.88x - 36[/tex]
If there exists profit, then P(x) has to be positive.
[tex]P(x) > 0 \\ 0.88x - 36 > 0 \\ 0.88x > 36 \\ x > \frac{36}{0.88} \\ x > 40.90909 \\ x = 41[/tex]
(x, y) → (x + 5, y − 4), what are the coordinates of point L′?
Answer: (5,-4)
Step-by-step explanation:
start at zero and go right five because of the adding of five
Then go down 4 because of the minus of zero.
Expand each binomial.
(2 x-y)⁴
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. (2x - y)⁴ is expanded as 16x⁴ − 32x²y + 24x²y² − 8xy³ + y⁴.
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 function be (2x - y)⁴
By using binomial expansion theorem, we get
(2x - y)⁴ = 16x⁴ − 32x²y + 24x²y² − 8xy³ + y⁴
(2x - y)⁴ is expanded as 16x⁴ − 32x²y + 24x²y² − 8xy³ + y⁴
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plss help me with this
Answer:
Below in bold.
Step-by-step explanation:
7x - 2 + 9x - 2 + 2(5x) = 2(5x + 5) + 2(4x)
7x + 9x + 10x - 4 = 10x + 10 + 8x
7x + 9x + 10x - 10x - 8x = 10 + 4
8x = 14
x = 14/8 = 7/4
So, ST = 5*7/4 + 5
= 55/4
= 13 3/4
Answer:
ST = 13.75
Step-by-step explanation:
The perimeter of a trapezium is calculated with the following formula:
P = a + b + c + d (a, b: bases, c, d: sides)
The perimeter of a rectangle is calculated with the following formula:
P= 2(a + b) (a: base, b: side)
we will write the following equation to find the length of ST
7x - 2 + 9x - 2 + 5x + 5x = 2(5x + 5 + 4x) first do the inside the parenthesis
2(9x + 5) = 18x + 10
Now,
7x - 2 + 9x - 2 + 5x + 5x = 18x + 10 add like terms on the left side of the equation
26x - 4 = 18x + 10 transfer like terms to the same side of the equation
26x - 18x = 10 + 4 add/subtract
8x = 14 divide both sides by 8
x = 1.75 replace x with this to calculate the lentgh of ST
5x + 5 = 5*(1.75) + 5 and ST = 13.75
Find the LCM of 2 and 6 using lists of multiples.
Answer:
6
Step-by-step explanation:
2,4,6,8,10
6,12,18,24
The least number that is on both lists is 6
Hilda has $210 worth of $10 and $12 stock shares. the numbers of $10 shares is 5 more than twice the number of $12 shares. how many of each does she have?
If Hilda has $210 worth of $10 and $12 stock shares and the number of $10 shares is 5 more than twice the number of $12 shares, then the number of $10 shares and $12 shares she has is equal to 15 and 5 respectively.
The number of $10 shares and $12 shares can be calculated by forming linear equations.
Consider the number of $10 shares to be x and the number of $12 shares to be y.
x = 2y + 5
10x + 12y = 210
Putting the value x = 2y + 5 in the second equation,
10x + 12y = 210
10 (2y + 5) + 12y = 210
20y + 50 + 12y = 210
32y + 50 = 210
32y = 210 - 50
32y = 160
y = 160 ÷ 32
y = 5
Putting y = 5 in the first equation,
x = 2y + 5
x = 2(5) + 5
x = 10 + 5
x = 15
Hence the number of $10 and $12 shares she has is calculated to be 15 and 5 respectively.
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What is the blue dot on
Calculate the percent increase or decrease between the starting and ending
quantities below. Round your answer to one decimal place.
1) Start: 3 End: 10
2) Start: 9 End: 20
3)Start: 100 End: 85
The percent increase or decrease of the starting and ending quantities below are:
233%122%-15%How to find the percent increase?The percent increase can be found by the formula:
= (End - Start) / Start x 100%
The percent increase of these quantities are:
= (10 - 3) / 3 x 100%
= 7 / 3 x 100%
= 2.3333 x 100%
= 233%
= (20 - 9) / 9 X 100%
= 11/9 x 100%
= 1.222 x 100%
= 122%
= (85 - 100) / 100 x 100%
= -15 / 100 x 100%
= -15%
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At a party store, packages of 12 noisemakers are $2 each. Packages of 9 party hats are $15 each. Mia wants to buy equal numbers of noisemakers and hats.
To get that many noisemakers and hats: How many packages of noisemakers does Mia need to buy?
How many packages of hats? Explain.