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anemone
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Prove $\sqrt[3]{43}<\sqrt[3]{9}+\sqrt[3]{3}<\sqrt[3]{44}$
my solution :anemone said:Prove $\sqrt[3]{43}<\sqrt[3]{9}+\sqrt[3]{3}<\sqrt[3]{44}----(1)$
it is easy :anemone said:Thanks Albert for participating...
I think you have to explicitly prove that $1.44<x=\sqrt[3]{3}<1.45 $ is true to complete your proof...:)
Albert said:it is easy :
$1.44^3=2.985984<3<1.45^3=3.048625$
I will use $AP>GP$ to prove $x>\dfrac{31}{9}$anemone said:Yes, it's easy and it's necessary. :D
My solution:
Let $x= \sqrt[3]{9} +\sqrt[3]{3}$, cube the inequality and rearrange, we have to prove $\dfrac{31}{9}\lt x \lt \dfrac{32}{9}$, but note that
$9(20^3)=72000\gt 41^3=68921$, this gives $\sqrt[3]{9} \gt \dfrac{41}{20}$ and
$3(25^3)=46875\gt 36^3=46656$, this gives $\sqrt[3]{3} \gt \dfrac{36}{25}$, so $x>\dfrac{41}{20}+\dfrac{36}{25}=3\dfrac{49}{100}>\dfrac{31}{9}$
Also,
$9(10^3)=9000< 21^3=9261$, this gives $\sqrt[3]{9} < \dfrac{21}{10}$ and
$3(20^3)=24000< 29^3=24389$, this gives $\sqrt[3]{3}<\dfrac{29}{20}$, so $x<\dfrac{21}{20}+\dfrac{29}{20}=2\dfrac{1}{2}<\dfrac{32}{9}$
so we're done.
The "Cube Root Challenge: Prove Inequality" is a mathematical problem in which one is asked to prove that the cube root of a number is always greater than or equal to the number itself. This challenge requires the use of algebraic manipulation and properties of cube roots to demonstrate the inequality.
This challenge is important because it helps to strengthen one's understanding of algebraic concepts and properties, specifically those related to cube roots. It also promotes critical thinking and problem-solving skills, which are essential in the field of science.
Some strategies for solving this challenge include manipulating the given equation using cube root properties, substituting different numbers to test the inequality, and using logical reasoning to support the proof.
Yes, for example, to prove that the cube root of 8 is greater than or equal to 8, we can start by writing the equation as ∛8 ≥ 8. Then, we can use the property that ∛a * b = ∛a * ∛b to rewrite the equation as ∛(2 * 4) ≥ 8. Next, we can simplify the cube root of the product of 2 and 4 to get ∛2 * ∛4 ≥ 8. Since ∛2 and ∛4 are both equal to 2, we can substitute 2 for each of them to get 2 * 2 ≥ 8, which is true. Therefore, we have proven that ∛8 ≥ 8.
This challenge can be applied in various fields, such as engineering, physics, and economics. For example, in engineering, one may need to prove the inequality between the root mean square (RMS) value and the average value of a signal. In physics, the challenge can be used to demonstrate the relationship between the average kinetic energy and the root mean square speed of gas molecules. In economics, the inequality can be applied to analyze the relationship between the average income and the root mean square deviation of income among a population.