To show of 5[SUP]√2 [SUP] >7, logically.

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In summary, the conversation discusses a homework problem where the goal is to prove that 5√2 > 7. The solution involves using the generalized binomial expansion to calculate 5^1.4 and showing that it is greater than 7. However, one participant suggests a simpler method using standard inequalities. Another participant suggests using logarithms to prove the inequality.
  • #1
ssd
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Stuck at the simple looking homework problem of my student:
To show 5√2 >7, logically.
 
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  • #2
Its ##5^{\sqrt2}>5^{1.4}##.

Now I suppose you aren't allowed to calculate powers other than integer powers, so you ll have to calculate 5^1.4 using the generalized binomial expansion ##(1+4)^{1.4}##. The first few terms up to 4^3 would be enough to show that ##5^{1.4}>7##.

https://en.wikipedia.org/wiki/Binomial_theorem#Generalisations
 
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  • #3
Delta² said:
Its ##5^{\sqrt2}>5^{1.4}##.

Now I suppose you aren't allowed to calculate powers other than integer powers, so you ll have to calculate 5^1.4 using the generalized binomial expansion ##(1+4)^{1.4}##. The first few terms up to 4^3 would be enough to show that ##5^{1.4}>7##.

https://en.wikipedia.org/wiki/Binomial_theorem#Generalisations

No need for that. If you want to prove ##5^{1.4} > 7##, then this is equivalent to proving ##(5^{14/10})^{10} > 7^{10}##. This should land you in the realm on the integers.
 
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Thanks for your response. So, we can start with 2>1.96 and calculate 514 and 710. But this is not the answer I was looking for. Actually this can be done using standard inequalities (it's my guess only).
 
  • #5
ssd said:
Thanks for your response. So, we can start with 2>1.96 and calculate 514 and 710. But this is not the answer I was looking for. Actually this can be done using standard inequalities (it's my guess only).
How about letting L = log base 2 and noticing:
L5√2 = √2L5 > 1.4L5 = 4(L5)/10 + L5 = (L54)/10 + L5 > .9 + L5 > ½ + L5 > L(1.4) + L5 = L(1.4•5) = L7.
 

What is the mathematical statement "To show of 5[SUP]√2 [SUP] >7, logically"?

The statement "To show of 5[SUP]√2 [SUP] >7, logically" means to prove that the value of 5[SUP]√2 [SUP] is greater than 7 using logical reasoning and mathematical principles.

Why is it important to prove that 5[SUP]√2 [SUP] >7?

Proving that 5[SUP]√2 [SUP] >7 is important because it helps establish the validity of mathematical principles and equations. It also allows us to confidently use this inequality in other mathematical calculations and applications.

What are the steps to logically prove 5[SUP]√2 [SUP] >7?

The steps to prove 5[SUP]√2 [SUP] >7 are: 1) Start by assuming that 5[SUP]√2 [SUP] is equal to 7. 2) Square both sides of the equation to get 25√2 = 49. 3) This leads to a contradiction, as 25√2 is not equal to 49. 4) Therefore, our initial assumption is false and the statement 5[SUP]√2 [SUP] >7 is true.

Can this inequality be proven using other methods?

Yes, there are multiple methods to prove the inequality 5[SUP]√2 [SUP] >7. One method is to use the fact that √2 is an irrational number and therefore, cannot be expressed as a ratio of two integers. Another method is to use the binomial theorem to expand 5[SUP]√2 [SUP] and compare it to the expansion of 7.

What are some real-life applications of this inequality?

This inequality can be applied in many real-life situations, such as calculating interest rates in financial investments, determining the angle of a ladder against a wall for stability, or analyzing the growth rate of a population. It also has applications in fields like engineering, physics, and computer science.

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