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Chapter 5: Problem 116

Which is greater \((\sqrt{n})^{\sqrt{n+1}}\) or \((\sqrt{n+1}) \sqrt{n}\) where \(n>8 ?\)

Short Answer

Expert verified
When \(n>8\), \((\sqrt{n+1}) \sqrt{n}\) is greater than \((\sqrt{n})^{\sqrt{n+1}}\) .

Step by step solution

01

Rewrite the expression

First, we need to rewrite the expression using the property \( \sqrt{n} = n^{1/2}\). This makes the expressions easier to compare. \n\((\sqrt{n})^{\sqrt{n+1}} = (n^{1/2})^{n^{1/2}+1/2}\) \n\((\sqrt{n+1})^{\sqrt{n}} = (n+1)^{1/2} * n^{1/2}\)
02

Simplify the expressions

Now, we must use the principle of power of a power for the first expression and apply the multiplication property of exponents for the second expression. \n\((n^{1/2})^{(n^{1/2}+1/2)} = n^{(n^{1/2}+1/2) * 1/2} = n^{n^{1/2}/2 + 1/4}\) \n\((n+1)^{1/2} * n^{1/2} = n^{1/2+(1/2)} = n^{2^{1/2}} = n\)
03

Compare the results

We now compare \((n^{n^{1/2}/2 + 1/4})\) with \(n\). It's worth noting that when \(n>8\), the value of \(n^{n^{1/2}/2 + 1/4}\) will always shrink faster than \(n\). Therefore, \((n^{n^{1/2}/2 + 1/4})\) is always less than \(n\) when \(n>8\).
04

Final comparison

Therefore, when \(n>8\), \((\sqrt{n+1}) \sqrt{n}\) is greater than \((\sqrt{n})^{\sqrt{n+1}}\) .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplification of Expressions
Simplifying expressions is a crucial step in solving mathematical problems. It involves rewriting expressions in a simpler or more comprehensible form without changing their value. In this exercise, we simplify the expressions by converting square roots into exponential form.
For example,
  • The expression \( \sqrt{n} \) can be rewritten as \( n^{1/2} \).
  • This translation makes handling complicated expressions much easier because exponents allow for straightforward algebraic manipulation.
By turning square roots into exponents, we can apply a wide range of arithmetic properties, like the power of a power property. This transformation is particularly helpful in comparing the size or magnitude of different expressions, as it provides a consistent and systematic approach.
Power of a Power Property
The power of a power property is a rule applied to simplify expressions that involve exponents. It states that when you raise a power to another power, you multiply the exponents. In mathematical terms:
  • \( (a^m)^n = a^{m \times n} \)
In the exercise solution, we use this property to simplify \( (n^{1/2})^{n^{1/2} + 1/2} \). Instead of dealing with the nested exponents directly, we simplify by multiplying:
  • The power to the outer exponent \( n^{1/2} \) by \( 1/2 \)
  • Resulting in \( n^{n^{1/2} / 2 + 1/4} \)
Using this property helps in handling complex expressions more systematically and aids in getting to the final answer quicker and more accurately.
Comparison of Expressions
When comparing expressions, especially with exponents, it鈥檚 essential to simplify them as much as possible first. After simplification using properties like the power of a power, we can easily compare expressions by analyzing their forms.
In our exercise:
  • We compare \( n^{n^{1/2} / 2 + 1/4} \) with \( n \).
  • The simplified expression \( n \) is straightforward and acts as our base for comparison.
Given that \( n > 8 \), we recognize that as \( n \) increases, \( n^{n^{1/2}/2 + 1/4} \) grows slower than \( n \). Thus, \( n^{n^{1/2}/2 + 1/4} \) will always be less than \( n \) for values of \( n \) over 8. By simplifying and utilizing properties, the comparison becomes clear and intuitive.

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Most popular questions from this chapter

(a) Show that \(\left(2^{3}\right)^{2} \neq 2^{(3)}\) . (b Are \(f(x)=\left(x^{x}\right)^{x}\) and \(g(x)=x^{\left(x^{\prime}\right)}\) the same function? Why or why not? (c) Find \(f^{\prime}(x)\) and \(g^{\prime}(x)\)

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