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otez555 [7]
2 years ago
6

Express (8 - 3sqrt(2))/(4 + 3sqrt(2)) in the form of a + b * sqrt(2) where a and b are integers.​

Mathematics
1 answer:
Naya [18.7K]2 years ago
3 0

Answer:   \boldsymbol{-25+18\sqrt{2}}

The expression is in the form a+b\sqrt{2} where a = -25 and b = 18

==========================================================

Work Shown:

Let x = 3\sqrt{2} and then square both sides

x = 3\sqrt{2}\\\\x^2 = \left(3\sqrt{2}\right)^2\\\\x^2 = \left(3\right)^2*\left(\sqrt{2}\right)^2\\\\x^2 = 9*2\\\\x^2 = 18\\\\

We'll use both x and x^2 later on.

---------------------

The denominator is in the form 4+x. We'll multiply top and bottom by 4-x to then use the difference of squares rule. This will allow us to eliminate the square root in the denominator.

\frac{8-3\sqrt{2}}{4+3\sqrt{2}}\\\\\frac{8-x}{4+x}\\\\\frac{(8-x)(4-x)}{(4+x)(4-x)}\\\\\frac{32-8x-4x+x^2}{16-x^2} \ \text{ difference of squares rule}\\\\\frac{32-12x+x^2}{16-x^2}\\\\

\frac{32-12*3\sqrt{2}+18}{16-18} \ \text{ plug in } x = 3\sqrt{2} \text{ and } x^2 = 18\\\\\frac{50-36\sqrt{2}}{-2}\\\\\frac{-2(-25+18\sqrt{2})}{-2}\\\\\boldsymbol{-25+18\sqrt{2}}

Therefore,

\frac{8-3\sqrt{2}}{4+3\sqrt{2}}=\boldsymbol{-25+18\sqrt{2}}

-------------------------------------

Checking the answer:

Use a calculator to find that,

\frac{8-3\sqrt{2}}{4+3\sqrt{2}} \approx 0.45584412271571 \\\\-25+18\sqrt{2} \approx 0.45584412271571

We get the same decimal approximation, which helps confirm the correct answer.

Or you could use the idea that if M = N, then M-N = 0 to subtract the original expression M and the final result N. You should get 0 or very close to it.

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Consider the probability that at least 93 out of 154 CDs will not be defective. Assume the probability that a given CD will not
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P(x =x) = nCx * p^x * (1 - p)^(n - x)

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3 years ago
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Answer:

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Step-by-step explanation:

<u>Surface Areas </u>

Is the sum of all the lateral areas of a given solid. We need to compute the total surface area of the given prism. It has 5 sides, two of them are equal (top and bottom areas) and the rest are rectangles.

Computing the top and bottom areas. They form a right triangle whose legs are 4.5 mm and 6 mm. The area of both triangles is

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The front area is a rectangle of dimensions 7.7 mm and 9 mm, thus

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The back left area is another rectangle of 4.5 mm by 9 mm

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Finally, the back right area is a rectangle of 6 mm by 9 mm

A_r=b.h=(6)(9)=54 \ mm^2

Thus, the total surface area of the prism is

A=A_t+A_f+A_l+A_r=27+67.5+40.5+54=189\ mm^2

\boxed{A=189\ mm^2}

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