Write the prime factorization of each number as a product of powers.
(a) 12^9 · 36^15 · 169^8
(b) 16^13 · 10^7 · 81^18

Each of the expressions can be rewritten using one of the properties of rational expressions as follows:

1. 49^1/2 * 3/5 = 3.21 (Power of  a power property)

2. 4^3/4 ÷ 4^1/4 = 4^3/4 -1/4 = 2.00 (Quotient of power property)

3. 27^4/3 ÷ 125^4/3 = 0.13 (Power of a quotient property)

4. 16^7/4 + 1/2 = 16^9/4 = 512 (Product of power property)

5. 36^3/2 + 64^3/2 = 216 + 512 = 728 (Power of product property)

6. 9^5/2 ÷ 25^5/2 = 243/3123 =0.78 (Power of a quotient property)

7. 81^5/6 - 1/3 = 81^1/2 = 9 (Quotient of powers property)

8. 64^4/3 ÷ 123^4/3 = 255.65/610.72 = 0.42 (Power of a quotient property)

9. x^13/10 y^2/5 * y^6/5  (Power of product property)

10. 4a^1/2 ÷ 2(4a^3/4) = 4a^-1/4 ÷ 2 = 2a^-1/4 (Quotient of powers property)

11. 27 * 2.99 x^6/5 - 1/5 = 80.91x (Quotient of powers property)

12. x^23/12 x^29/12 y^2     (Power of product property)

A rational expression is a mathematical representation that aims at indicating the ratios of two different or similar polynomial values. The alphabets x and y are often used in such expressions.

It is also possible to rewrite rational expressions using some laid down principles. The given expressions have been rewritten with the relevant rational rules.

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Answer:

Step-by-step explanation:The width would be 2 the same for the height

The expression [tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}[/tex] is equivalent to 10 for  [tex]x = 10^{(8/15)}[/tex].

In mathematics, an expοnent (alsο called pοwer οr index) is a shοrthand nοtatiοn used tο represent the repeated multiplicatiοn οf a number by itself.

We can simplify the given expressiοn using the prοperties οf expοnents and rοοts.

First, we can simplify the expression inside the first square root as:

√√x1⁰ = [tex]\rm \sqrt{(x^{10})^{(1/4)} }= (x^{10})^{(1/8)} = x^{(10/8)} = x^{(5/4)}[/tex]

Next, we can simplify the expression inside the second square root as:

[tex]\sqrt{(\sqrt{x\³})} = \sqrt{(x^3)^{(1/4) }}= (x^3)^{(1/8)} = x^{(3/8)[/tex]

Therefore, the entire expression becomes:

[tex]\sqrt{(\sqrt{x1\⁰)}} (\sqrt{(\sqrt{x\³})} = x^{(5/4)} * x^{(3/8) }= x^{(15/8)}[/tex]

Now, we need to find the value of x that makes this expression equal to 10.

[tex]x^{(15/8) }= 10[/tex]

Taking the eighth power of both sides:

[tex]x^{15} = 10^8[/tex]

Taking the 15th root of both sides:

[tex]x = (10^8)^{(1/15)}[/tex]

[tex]x = 10^{(8/15)}[/tex]

the equivalent expression is:

[tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}= x^{(5/4) }* x^{(3/8)} = x^{(15/8)} = (10^{(8/15)})^{(15/8)} = 10[/tex]

Therefore, the expression [tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}[/tex] is equivalent to 10 for [tex]x = 10^{(8/15)}[/tex].

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Monica is not correct. Monica will earn $13.50 in 1 year while Paul will earn $13.20 in 1 year.

Simple interest is a linear function of the amount deposited, the interest rate and the number of years.

Simple interest = amount deposited x interest rate x time

Interest earned by Monica = $400 x 0.033 x 1 = $13.20

Interest earned by Paul = $500 x 0.027 x 1 = $13.50

Difference in the interest = $13.50 - $13.20 = $0.30

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The required expression represents the value in dollars of the coin n years after Hector found is [tex]$c(n) = 190 \times (1.1)^n$[/tex].

Let's call the percentage increase in value each year "r". We know that after one year, the coin was worth $209, which is 209/190 = 1.1 times what it was worth when Hector found it. So we can set up an equation:

[tex]$190 \times (1 + r)^1 = 209$[/tex]

Simplifying this equation,

[tex]$1 + r = \sqrt[1]{\frac{209}{190}}$[/tex]

1 + r = 1.1

r = 0.1

So the percentage increase in value each year is 10%. To find the value of the coin "n" years after Hector found it, we can use the formula:

[tex]$c(n) = 190 \times (1 + r)^n$[/tex]

Substituting the value we found for "r", we get:

[tex]$c(n) = 190 \times (1.1)^n$[/tex]

So the expression for the value of the coin "n" years after Hector found it is:

[tex]$c(n) = 190 \times (1.1)^n$[/tex]

Thus, required expression represents the value in dollars of the coin n years after Hector found is [tex]$c(n) = 190 \times (1.1)^n$[/tex].

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Check the picture below.

[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ S = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ n=4 \end{cases}\implies S=180(4-2)\implies S=360 \\\\[-0.35em] ~\dotfill\\\\ (40)+(2x)+( ~~ 360-(x+100) ~~ )+(x)~~ = ~~360 \\\\\\ (40)+(2x)+( ~~ 360-x-100 ~~ )+(x)~~ = ~~360\implies 2x+300=360 \\\\\\ 2x=60\implies x=\cfrac{60}{2}\implies \boxed{x=30}[/tex]

Answer:

  80°

Step-by-step explanation:

You want the larger of two acute angles in a right triangle if their relationship is cos(8x°) = sin(2x-10°).

The cosine of an acute angle is equal to the sine of its complement. The given relation between the angles means ...

  (8x) +(2x -10) = 90 . . . . . the angles are complementary

  10x = 100 . . . . . . . . . . . add 10

  x = 10 . . . . . . . . . . . . . divide by 10

The two angles are 8·10° = 80° and (2·10 -10)° = 10°.

The larger of the two acute angles is 80°.

Answer:

Step-by-step explanation:

You want to know the parts of the composition of transformations that will map triangle A to A', then to A''. You also want to know what single transformation accomplishes the same thing.

The long side of triangle A points "south". In triangle A', it points "east". To transform a segment pointing south to one pointing east requires a rotation of +90°, or 90° CCW.

The long side of triangle A' lies along the line y=1. In triangle A'', it lies on the line y=-1. The line of reflection is halfway between these values, at ...

  y = (1 +(-1))/2 = 0/2 = 0

The line of reflection is y=0, the x-axis.

We know the transformation involves a reflection. The line of reflection can be found by finding the midpoints of the original and final vertices of the triangle.

The midpoint of the largest acute angle's vertex is ...

  ((2, -3) +(2, -3))/2 = (2, -3)

The midpoint of the right angle vertex is ...

  ((0, -3) +(2, -1))/2 = (1, -2)

The line through these two points is ...

  x +y = -1

The transformation of A to A'' is a reflection in the line x+y=-1.

The ratio of the number of squares to the number of triangles in simplest form is 5:2. The number of triangles that need to be added to make the ratio of the number of squares to the number of triangles 1 : 1 is 3.

What is the ratio?

A fraction that demonstrates how several times one quantity is of the other is the ratio of two quantities of the same kind and in the same units. Ratio is represented by the symbol ":". The expression a/b, which originally stood for the ratio of two measurements a and b (b≠ 0), is used.

Given that the number of squares are 5 and the number of triangles are 2.

The ratio of the number of squares to the number of triangles in simplest form is 5 : 2.

Assume that x number of triangles that needs to be added to make the ratio of the number of squares to the number of triangles 1 : 1.

The equation is:

5:(2+x) = 1:1

5/(2+x) = 1/1

Cross multiply both sides:

5 = 2 + x

x = 5 - 2

x = 3

3 triangles need to be added to make the ratio of the number of squares to the number of triangles 1 : 1

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a) The probability that it will not rain on the homecoming parade in Amina's first year is 0.84. b) The probability that it will not rain on any of the homecoming parades during Amina's four years at the school is 0.49. c) The probability that it will rain on Amina's parade at least once during her four years of attendance is 0.51.

We can solve this problem as:
a) This can be calculated by subtracting the probability of rain during the parade (0.16) from 1.

So, the probability is,

1 - 0.16 = 0.84.

b) This can be calculated by taking the probability of no rain (0.84) and multiplying it by itself four times.

So, the probability is,

0.84^4 = 0.49.

c) This can be calculated by subtracting the probability of no rain (0.49) from 1.

So, the probability is,

1 - 0.49 = 0.51.

In conclusion, the probability that it will not rain on the homecoming parade in Amina's first year is 0.84, the probability that it will not rain on any of the homecoming parades during Amina's four years at the school is 0.49, and the probability that it will rain on Amina's parade at least once during her four years of attendance is 0.51.

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This modification will reduce the amount he needs to borrow from the lender and thus lower his monthly mortgage payments.

The down payment is the amount of money that the buyer pays upfront towards the purchase price of the property. The more significant the down payment, the less money the buyer has to borrow from the lender, and this can have a significant impact on the monthly mortgage payments.

In the case of Nick, he has saved up 35,000 for a down payment, which is approximately 18% of the purchase price of the house. If he decides to go with the 20% down payment option, he will need to pay 38,400 upfront, which is 20% of 192,000. The remaining 153,600 will be the amount he borrows from the lender.

Now, let's assume that Nick has taken a 30-year fixed-rate mortgage at 6.25% interest. If he decides to pay 20% down payment, his monthly mortgage payments would be approximately 940. However, Nick wants to spend no more than 30% of his income on his mortgage payment, which is 997.50 per month (30% of 3,325). Therefore, the modification that can be made to this loan to make it a viable option is to reduce the loan amount.

If Nick reduces the loan amount to 153,000, his monthly mortgage payment will be approximately 910, which is within his budget. To achieve this, he will need to increase his down payment to 39,000, which is 20.3% of the purchase price. This modification will reduce the amount he needs to borrow from the lender and thus lower his monthly mortgage payments.

In summary, a larger down payment reduces the amount borrowed from the lender and, consequently, lowers the monthly mortgage payments. To make the loan option viable, Nick needs to increase his down payment to reduce the loan amount and meet his budget.

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each person would have 2 feet and two inches of tape

The square pizza has an area of about 42.06 square inches more than the circular pizza, meaning that the square pizza is approximately 28% larger than the circular pizza.

The area of a circle with a diameter of 14 inches is (π/4) x (14 inches)², which is approximately 153.94 square inches.

The area of a square with sides measuring 14 inches is 14 inches x 14 inches, which is 196 square inches.

Therefore, the circular pizza has an area that is approximately 42.06 square inches smaller than the square pizza. In other words, the square pizza is approximately 28% larger than the circular pizza.

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The values of a and b are a = 0 and b = -3, and the quadratic equation of the curve is y = (x - 8)² - 3.

What is a quadratic function?

A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

In this case, the vertex is (8, -3).

So we have -

y = a(x - 8)² - 3

To find the value of a, we need one more point on the parabola.

Let's use the fact that the curve is symmetric about the vertex, so there must be another point with the same y-value, but reflected across the vertex.

Since the vertex is at (8, -3), the reflection would be at (8 - 2a, -3).

So we have -

-3 = a(8 - 2a - 8)² - 3

Simplifying -

0 = a(2a)²

0 = 4a³

So either a = 0 or a is undefined (i.e. the parabola is a horizontal line). But we know the curve has a minimum, so it can't be a horizontal line.

Therefore, a must be 0.

Substituting this into our equation, we get -

y = 0(x - 8)² - 3

y = (x - 8)² - 3

Therefore, a = 0 and b = -3, and the equation is y = (x - 8)² - 3.

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Answer:

y > -5

Step-by-step explanation:

Answer: We can solve this problem using trigonometry and basic geometry. Let's first draw a diagram:

               A (top of house)

              /|

             / | h

            /  |

           /   |

          /θ1  |

         /     |

        /      |

       /       |

      /        |

     /         |

    /          |

   /           |

  /            |

 /             |

B (top of column)  

 | d          

 |            

 |

 |

C (bottom of column)

We are given that angle theta 1 (θ1) is 45 degrees, angle theta 2 (θ2) is 60 degrees, and the height of the column BC is 95.62 meters. We want to find the height of the house AB, which we'll call "h".

First, we can use trigonometry to find the length of segment CD. Since we know angle θ2 and the length of BC, we can use the tangent function:

tan(θ2) = opposite / adjacent

tan(60°) = d / 95.62

d = 95.62 * tan(60°)

d ≈ 165.30 meters

Now we can use the fact that angles ABD and CBD are complementary to find the length of segment AD:

tan(θ1) = opposite / adjacent

tan(45°) = h / (d + 165.30)

h = (d + 165.30) * tan(45°)

h ≈ 165.30 meters

Therefore, the height of the house is approximately 165.30 meters.

Step-by-step explanation:

Cost of materials for each boy's costume = 5 * 4 * 4 + 5 * 3 * $.50 = 80 + 7.5 = $87.50 Cost of materials for each girl's costume = 6 * 5 * 4 + 6 * 2 * $2 = 120 + 24 = $144

Let A = [5; 6], B = [4; 5], C = [3; 2], D = [4; 2], E = [$.50; $2].

Cost of materials for each costume = A * B * D + A * C * E

Cost of materials for each costume = [5; 6] * [4; 5] * [4; 2] + [5; 6] * [3; 2] * [$.50; $2]

Cost of materials for each costume = [80; 120] + [15; 24]

Cost of materials for each costume = [95; 144]

Total cost of materials for each costume = $239

Cost of materials for each boy's costume = 5 * 4 * 4 + 5 * 3 * $.50 = 80 + 7.5 = $87.50

Cost of materials for each girl's costume = 6 * 5 * 4 + 6 * 2 * $2 = 120 + 24 = $144

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a.sin(3θ)cos(θ) - cos(3θ)sin(θ) = sin(4θ) - sin(2θ) = sin(4θ - 2θ) = sin(2θ)

b.2θ = 0, π, 2π, 3π,...

To solve this equation, first use addition or subtraction to simplify. In this case, it is best to add the two terms together and use the trigonometric identity sin(A)cos(B) + cos(A)sin(B) = sin(A+B). Applying this, we can rewrite the equation as:

sin(3θ)cos(θ) - cos(3θ)sin(θ) = sin(4θ) - sin(2θ) = sin(4θ - 2θ) = sin(2θ)

Now, to find all solutions of the simplified equation, use the fact that sin(2θ) = 0 when 2θ = 0, π, 2π, 3π,... Using this, the solutions of the equation are:

2θ = 0, π, 2π, 3π,...

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The exponential function of the line that goes through points (0, 20) and (2, 180) is y = 20 × 3ˣ.

To find the exponential function of a line that goes through two points, we need to use the general form of the exponential function:

y = a × bˣ

where a is the initial value, b is the base, and x is the independent variable (usually time).

We can use the two points given to find the values of a and b.

Let's start with the point (0, 20):

y = a × bˣ

Plug in x = 0 and y = 20

20 = a ×  b⁰

20 = a ×  1

a = 20

Next, let's use the point (2, 180):

y = a × bˣ

Plug in x = 2 and y = 180

180 = 20 × b²

b² = 180/20

b² = 9

b = 3

Now, plug these values into y = a × bˣ

Therefore, the exponential function y = 20 × 3ˣ

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[tex]\tan^2(\theta )+3\tan(\theta )=0\implies \tan(\theta )[\tan(\theta )+3]=0 \\\\[-0.35em] ~\dotfill\\\\ \tan(\theta )=0\implies \theta =\tan^{-1}(0)\implies \theta = \begin{cases} 90^o\\ 270^o \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \tan(\theta )+3\implies \tan(\theta )=-3\implies \theta =\tan^{-1}(-3)\implies \theta \approx \begin{cases} \stackrel{ II~Quadrant }{108.43^o}\\ \stackrel{ IV~Quadrant }{288.43^o} \end{cases}[/tex]

We will thus require 62 cubοid bricks tο create the wall, tο the clοsest whοle brick.

A nοrmal brick is a cubοid, a three-dimensiοnal structure with six rectangular faces.

We need tο have at minimum 42/18 = 2.33 rοws mοre bricks because the wall is 42 inches tall, and each brick is 18 inches tall. We require three rοws οf bricks since we can't have a half rοw.

We require 144/7 = 20.57 bricks each rοw because the wall is 144 inches lοng and each brick is 7 inches lοng. There must be 21 bricks each rοw because we cannοt have half bricks.

The first rοw requires 21 whοle bricks. If a brick is sliced in half, the οther half can be used οn the οther end οf the identical rοw.

21 whοle bricks are alsο required fοr the secοnd rοw. Fοr the remainder οf the rοw, we need οnly 20 full bricks since we can utilise the half-brick we cut during the initial rοw just at start οf the secοnd rοw.

The third rοw requires 21 whοle bricks. We have a spare half-brick that we may utilize at the cοnclusiοn οf the third rοw since we utilized οne in the first rοw.

Hence, 62 cοmplete bricks are needed, which equals 21 + 20 + 21.

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The cup will hold more ice cream than cone since volume of cup is 169.6 cubic inches which is greater than volume of 25.1 cubic inches.

The volume of a cone is given by the formula:

V= 1/3.π[tex]r^{2}[/tex]h

where r is the radius of the circular base and h is the height.

The volume of a cylinder is given by the formula:

V= π[tex]r^{2}[/tex]h

where r is the radius of the circular base and h is the height.

Assuming that the height of both the cone and the cup is the same, we can compare their volumes by comparing their radii.

For the cone, the radius r is equal to half the diameter of the circular base, which is equal to w/2 = 2 in.

The height of the cone is equal to y, which is 6 in.

V= 1/3.π[tex]r^{2}[/tex]h= 1/3.π[tex]2^{2}[/tex]6

For the cup, the radius r is equal to x/2 = 3

The height of the cup is equal to y, which is also 6 in.

Using these values, we can calculate the volume of the cup as:

[tex]V_cup[/tex]= π[tex]r^{2}[/tex]h=π[tex]3^{2}[/tex]6

Therefore, the cup will hold more ice cream than the cone.

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Answer:

Step-by-step explanation:u better not be him cause im him

Simple Interest = (Principal x Rate x Time)

where:

Principal = R500 (the amount invested)

Rate = 12% per annum = 0.12 (expressed as a decimal)

Time = 4 years

So, the simple interest earned by John over 4 years is:

Simple Interest = (500 x 0.12 x 4) = R240

To find the total amount of money John will have at the end of 4 years, we need to add the simple interest to the principal:

Total amount = Principal + Simple Interest = 500 + 240 = R740

Therefore, John will have R740 at the end of 4 years.

Answer:

The total amount he has at the end is ₹740.

Step-by-step explanation:

Simple Interest = [tex]\frac{PRT}{100}[/tex]

                          =  [tex]\frac{500 * 12 * 4}{100}[/tex] = ₹240

Total amount he has at the end of 4 years = Principal + Interest
                                                                       = ₹500 + ₹240

                                                                       = ₹740

Answer: Bonato

Step-by-step explanation:

The domain of function f is (-∞, 10) U (10, ∞) or {x|x ≠ 10}.

The range of function f is (-∞, -3) U (-3, ∞) or {y|y ≠ 3}.

The inverse of this function is f⁻¹ = (10x - 4)/(3 + x).

The domain of function f⁻¹ is (-∞, -3) U (-3, ∞) or {x|x ≠ 3}.

The range of function f⁻¹ is (-∞, 10) U (10, ∞) or {y|y ≠ 10}.

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In order to determine the inverse of any function, you should swap both the input value (x-value) and output value (y-value) as follows;

f(x) = y = (3x + 4)/(10 - x)

x = (3y + 4)/(10 - y)

10x - yx = 3y + 4

3y + yx = 10x - 4

y(3 + x) = 10x - 4

f⁻¹ = (10x - 4)/(3 + x)

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The point which is a solution to the equation 2x - 7y = 7 as required to be determined is; (0, -1).

As evident from the task content; the point which is a solution to the given equation is to be determined.

For a given equation; the y-intercept represents a solution to the equation.

On this note, if x = 0; therefore, we have that;

2 (0) - 7y = 7

y = 7 / -7

y = -1.

Ultimately, the point which is a solution to the equation is; (0, -1).

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Answer:

Step-by-step explanation: It would help if you added the picture but its all good.

The total number of times you would expect the number 7 to appear is:1000 × 1/6 = approximately 143 times.

If you were to roll two dice 1000 times, the number of times you would expect the number 7 to appear is approximately 143 times.The possible outcomes when two dice are rolled are 6 × 6 = 36. The probability of obtaining a total of 7 when two dice are rolled is:6/36, which reduces to 1/6.

The total number of times you would expect the number 7 to appear is:1000 × 1/6 = approximately 143 times.

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