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olya-2409 [2.1K]
3 years ago
12

Explain why the cube root of 64 is rational and the cube root of 1 2 is not.

Mathematics
1 answer:
Ira Lisetskai [31]3 years ago
7 0

\sqrt[3]{64}=4 so it's obvious, I guess :)

Now, let's prove \sqrt[3]{12} is not rational number.

Proof by contradiction.

Let assume \sqrt[3]{12} is a rational number. Therefore it can be expressed as a fraction  \dfrac{a}{b} where a,b\in\mathbb{Z} and \text{gcd}(a,b)=1.

\sqrt[3]{12}=\dfrac{a}{b}\\\\12=\dfrac{a^3}{b^3}\\\\a^3=12b^3

12b^3 is an even number, so a^3 must also be an even number, and therefore also a must be an even number. So, we can say that a=2k, where k\in\mathbb{Z}.

(2k)^3=12b^3\\\\8k^3=12b^3\\\\2k^3=3b^3

Since 2k^3 is an even number, then also 3b^3 must be an even number. 3 is odd, so for 3b^3 to be an even number, b^3 must be an even number, and therefore b is an even number.

But if both a and b are even numbers, then it contradicts our earlier assumption that \text{gcd}(a,b)=1. Therefore \sqrt[3]{12} is not a rational number.

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Can you help me pleeeeaaassee??
melamori03 [73]

Error 1: DT / TS × CT / TR which is the first error. We can fix the error by writing the correct equation RT / TD = ST / TC, Error 2: The second error is 7 / 16 × (x - 1) / 14 and we can fix the error by writing the equation 14 / 7 = 16 / (x - 1), Error 3:The third error is the value of x and we can find the correct value of x from the equation 14 / 7 = 16 / (x - 1) and the value of x is 9.

Given: The diagram is given and we need to find the errors and then fix them. Also ΔTSR ≈ ΔTCD

Let's solve the given question:

Given that ΔTSR ≈ ΔTCD

So we know by the properties of the similarity that if two triangles are similar then the ratio of their corresponding sides is equal.

So, ΔTSR ≈ ΔTCD

=> RT / TD = ST / TC

=> 14 / 7 = 16 / (x - 1)

In the question, we can observe that the given side ratio is DT / TS × CT / TR which is the first error. We can fix the error by writing the correct equation RT / TD = ST / TC.

The second error is 7 / 16 × (x - 1) / 14 and we can fix the error by writing the equation 14 / 7 = 16 / (x - 1).

The third error is the value of x.

We can find the correct value of x from the given equation:

14 / 7 = 16 / (x - 1)

=> 2 = 16 / (x - 1)

Multiplying both sides by (x - 1):

(x - 1) × 2 = 16 / (x - 1) × (x - 1)

=> 2(x - 1) = 16

Multiplying both sides by 1 / 2:

2(x - 1) × 1 / 2= 16 × 1 / 2

=> x - 1 = 8

Adding 1 on both sides:

x - 1 + 1 = 8 + 1

x = 9

Therefore x = 9.

Hence the errors are:

Error 1: DT / TS × CT / TR which is the first error. We can fix the error by writing the correct equation RT / TD = ST / TC

Error 2: The second error is 7 / 16 × (x - 1) / 14 and we can fix the error by writing the equation 14 / 7 = 16 / (x - 1).

Error 3:The third error is the value of x and we can find the correct value of x from the equation 14 / 7 = 16 / (x - 1) and the value of x is 9.

Know more about "similar triangles" here: brainly.com/question/14366937

#SPJ9

8 0
1 year ago
△XYZ is mapped to △X'Y'Z' using the rules (x, y)→(x+5, y−3) followed by (x, y)→(−x, −y) .
Yuri [45]

Given that △XYZ is mapped to △X'Y'Z' using the rules (x, y)→(x+5, y−3) followed by (x, y)→(−x, −y) .

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We know that (x,y) => (-x,-y) type operation represents rotation about origin so rule (x, y)→(−x, −y) will cause rotation.

So combining both results and comparing with given choices. we find that only 1st choice "△XYZ is congruent to △X'Y'Z' because the rules represent a translation followed by a rotation, which is a sequence of rigid motions."

is correct.

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3 years ago
The populations and land areas of four cities in Texas are shown.
dybincka [34]
We are given a table showing the population and the area of different cities, A, B, C and D. To calculate the population density, divide the population over the total area of the city. The following statements are true:

<span>--The population density of City B is greater than that of City C.

--City D has the lowest population density of the four cities.

</span><span>--The population density for City B can be found using the ratio 48,592 : 26.</span><span>
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The value of x would be 7
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