Solve for z.
[tex]z {}^{4} - 65z {}^{2} + 64 = 0[/tex]

We have to prove that [tex]\sqrt{2}+\sqrt{5}[/tex] is irrational. We can prove this statement by contradiction.

Let us assume that [tex]\sqrt{2}+\sqrt{5}[/tex] is a rational number. Therefore, we can express:

[tex]a=\sqrt{2}+\sqrt{5}[/tex]

Let us represent this equation as:

[tex]a-\sqrt{2}=\sqrt{5}[/tex]

Upon squaring both the sides:

[tex](a-\sqrt{2})^{2}=(\sqrt{5})^{2}\\a^{2}+2-2\sqrt{2}a=5\\a^{2}-2\sqrt{2}a=3\\\sqrt{2}=\frac{a^{2}-3}{2a}[/tex]

Since a has been assumed to be rational, therefore, [tex]\frac{a^{2}-3}{2a}[/tex] must as well be rational.

But we know that [tex]\sqrt{2}[/tex] is irrational, therefore, from equation [tex]\sqrt{2}=\frac{a^{2}-3}{2a}[/tex] the expression [tex]\frac{a^{2}-3}{2a}[/tex] must be irrational, which contradicts with our claim.

Therefore, by contradiction,  [tex]\sqrt{2}+\sqrt{5}[/tex] is irrational.

Step-by-step explanation:

Let us assume [tex]\bf \sqrt{2}+\sqrt{5} [/tex] as rational.

[tex]\therefore \: \:\bf \sqrt{2}+\sqrt{5} = \dfrac{a}{b}[/tex]

[tex]\implies\:\:\bf \sqrt{5} = \dfrac{a}{b} - \sqrt{2} [/tex]

[tex]\implies\:\:\bf(\sqrt{5})^{2} = \bigg( \dfrac{a}{b} -\sqrt{2}\bigg)^{2} [/tex]

[tex]\implies\:\:\bf 5 = \dfrac{a^{2} }{b^{2}} - 2 \bigg(\dfrac{a}{b}\bigg) (\sqrt{2}) + (\sqrt{2})^{2}[/tex]

[tex]\implies\:\:\bf 5 = \dfrac{a^{2} }{b^{2}} - 2 \sqrt{2} \bigg( \dfrac{a}{b} \bigg) + 2[/tex]

[tex]\implies\:\:\bf 5-2 = \dfrac{a^{2}}{b^{2}}- 2 \sqrt{2} \bigg( \dfrac{a}{b} \bigg)[/tex]

[tex]\implies\:\:\bf 3 = \dfrac{a^{2}}{b^{2}}- 2 \sqrt{2} \bigg( \dfrac{a}{b} \bigg)[/tex]

[tex]\implies\:\:\bf 2 \sqrt{2} \bigg( \dfrac{a}{b} \bigg)= \dfrac{a^{2}}{b^{2}} -3[/tex]

[tex]\implies\:\:\bf 2 \sqrt{2} \bigg( \dfrac{a}{b} \bigg)=\dfrac{a^{2}-3b^{2}}{b^{2}}[/tex]

[tex]\implies\:\:\bf 2 \sqrt{2} = \bigg( \dfrac{a^{2}-3b^{2}}{b^{2}} \bigg) \bigg( \dfrac{b}{a}\bigg) [/tex]

[tex]\implies\:\:\bf \sqrt{2} =\dfrac{a^{2}-3b^{2} }{2ab}[/tex]

[tex]\textsf Hence, our assumption is wrong [/tex]

[tex] \therefore \:\:\sf {{\bf{\sqrt{2}+\sqrt{5}}} is irrational }[/tex]

[tex]\mathfrak\purple{{\purple{\bf{H}}}ence\: {\purple{\bf{P}}}roved}[/tex]

Answer:

Option D

Step-by-step explanation:

Given that John drives 257 miles and uses 9 gallons of gas.

we can find unit rate as

Per galon mileage given = 257/9 = 28

ANd units are miles/gallon

100 miles per gallon is very far from 28 hence incorrect

6 miles per gallon is very low

30 is right but unit is wrong

Hence option D is the right answer.

Answer: ITS NOT 1/6, ITS 1/12

Step-by-step explanation:

I took the test :)

Answer:  The correct option is

(B) a 90° clockwise rotation.

Step-by-step explanation:  Given that the co-ordinates of the vertices of figure FGH are  F(1, 3), G(–1, 2) and H(2, 1).

The figure FGH is rotated using origin as the center of rotation and the the co-ordinates of the vertices of image figure F'G'H' are  F'(3, -1), G'(2, 1) and H'(1, -2).

We are to select the type of rotation that could have taken place.

The transformation from the vertices of FGH to the vertices of image F'G'H' is

F(1, 3)  ⇒   F'(3, -1),

G(-1, 2)  ⇒   G'(2, 1),

H(2, 1)  ⇒   H'(-1, 2).

Therefore, the transformation rule is  (x, y)  ⇒   (y, -x). This gives the rotation about origin through an angle of 90°.

Thus, the rotation of 90° clockwise taken place from figure FGH to image F'G'H'.

Option (B) is correct.

Answer:

4 batches

Step-by-step explanation:

Let

x -----> the number of cups of apples

y -----> the number of cups of mangoes

z -----> the number of cups of Blueberries

w -----> the number of cups of Blackberries

we know that

x/y=3/1 -----> equation A

x/z=3/2 ----> equation B

x/w=3/1 ----> equation C

x+w=16 ----> equation D

Solve the system of equations C and D

x=3w ----> equation C

x+w=16 ----> equation D

Substitute equation C in equation d and solve for w

3w+w=16

4w=16

w=4 cups of Blackberries

Remember that

For 1 batch of the recipe are used 1 cup of Blackberries

so

For 4 cups of Blackberries

The number of batches are

(1)*4=4 batches

Answer:

the answer is b

Step-by-step explanation:

it is

If triangle ABC is a right triangle with ∠C = 90°, then angles A and B must be complementary.

If triangle ABC is a right triangle with ∠C = 90°, then angles A and B must be complementary. This means that the sum of angles A and B will be equal to 90°. Therefore, the correct answer is option b.

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The true statement about angles A and B is that a. Angles A and B are complementary.

Since triangle ABC is a right triangle with m∠C = 90°, we know that m∠A + m∠B = 90°.

This means that the sum of the measures of angles A and B is 90 degrees.

So, angles A and B are complementary angles.

In other words, if two angles are complementary, then the sum of their measures is equal to 90 degrees.

Read more about angles at

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Question

if aabc is a triangle with mlc = 90, which of the following must be true about angles a and b?

Angles A and B are complementary.

Angle B is greater than angle A.

Angles A and B are supplementary

Angle A is greater than angle B.

Answer:

C. commutative property

Step-by-step explanation:

Answer:

140

Step-by-step explanation:

that's the answer on ttm

Answer: (8,-6)

Step-by-step explanation:

the coordinates are  (3,-2)

(x+5, y-4)

(3+5,-2-4)

(8,-6)

Answer:

5.25

Step-by-step explanation:

It takes approximately 60/7 seconds for the population to grow by one person.

To find out how many seconds it takes for the population to grow by one person, we need to determine the difference between the rate of births and the rate of deaths. The rate of population growth per second can be calculated by subtracting the rate of deaths per second from the rate of births per second. In this case, a person is born every 5 seconds and a person dies every 12 seconds. So, the rate of population growth per second is 1/5 - 1/12 = 12/60 - 5/60 = 7/60.

Therefore, it takes approximately 60/7 seconds for the population to grow by one person.

Applying the multiplication principle in probability, there are eight different combinations possible when choosing between bus or train, full fare or concession, morning or afternoon for your travel.

Your question is asking how many different combinations of travel mode, fare type, and time of day are possible under given conditions. For this situation, we can apply simple probability rules with concept of multiplication principle, which states that if there are n ways to do something, and m ways to do another, then there are n * m ways to do both.

In this case, you have two choices for each category: mode of transportation (bus or train), fare type (full fare or concession), and time of day (morning or afternoon). The combination is simply 2 (mode of transport) * 2 (fare type) * 2 (time of day) = 8. So, there are eight possible outcomes.

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Answer:

the answer is B

Step-by-step explanation:

The content of care package would be a board game and several snack packs.

Given is the weight of board game = 4 pounds.

The weight of one snack pack = [tex] \frac{1}{2} \;pounds = 0.5 \;pounds [/tex]

The total weight would be less than 25 pounds.

Solving for part A :

Let 'x' represents the number of snack packs.

So the weight of all snack packs would be 0.5x pounds.

Now weight of care pack = weight of board game + weight of all snack packs.

weight of care pack = 4 + 0.5x

But total weight must be less than 25 pounds.

So the inequality would be: 4 + 0.5x < 25

Solving for part B :

Solving the inequality 4 + 0.5x < 25

⇒ 4 + 0.5x - 4 < 25 - 4

⇒ 0.5x < 21

⇒ x < 42

Solving for part C :

Since x < 42 and x is an integer number, so x = 41.

She can include maximum of 41 snack packs in the care package.