If \(a^b = b^a\), then what is \(\frac{a \times (\frac{a}{b}) ^{ \frac{a}{b} } }{ (a)^{ \frac{a}{b} } }\) equal to?
Understanding the Problem
We are given an equation relating two variables, \(a^b = b^a\), and asked to find the value of a specific mathematical expression involving these variables:
\(E = \frac{a \times \left(\frac{a}{b}\right)^{ \frac{a}{b} } }{ a^{ \frac{a}{b} } }\)Our goal is to simplify this expression using the given condition.
Simplifying the Expression
Let's break down the simplification process step-by-step:
- Rewrite the expression using the properties of exponents. Specifically, we can rewrite the division as multiplication by a negative exponent: \(E = a \times \left(\frac{a}{b}\right)^{ \frac{a}{b} } \times a^{ - \frac{a}{b} }\)
- Group terms with the same base (\(a\) and \(b\)). Recall that \(\left(\frac{a}{b}\right)^{ \frac{a}{b} } = \frac{a^{ \frac{a}{b} }}{b^{ \frac{a}{b} }}\). Substituting this in the original expression gives: \(E = \frac{a \times \frac{a^{ \frac{a}{b} }}{b^{ \frac{a}{b} }} }{ a^{ \frac{a}{b} } }\) Now, simplify the fraction: \(E = a \times \frac{a^{ \frac{a}{b} }}{b^{ \frac{a}{b} }} \times \frac{1}{a^{ \frac{a}{b} }}\)
- Cancel out common factors. The term \(a^{ \frac{a}{b} }\) appears in both the numerator and the denominator: \(E = a \times \frac{1}{b^{ \frac{a}{b} }}\) \(E = \frac{a}{ b^{ \frac{a}{b} } }\)
Using the Condition \(a^b = b^a\)
Now we need to use the given condition \(a^b = b^a\) to simplify the expression \(E = \frac{a}{ b^{ \frac{a}{b} } }\).
Let's manipulate the condition:
- Start with \(a^b = b^a\).
- Raise both sides to the power of \(\frac{1}{b}\): \((a^b)^{\frac{1}{b}} = (b^a)^{\frac{1}{b}}\)
- Using the exponent rule \((x^m)^n = x^{mn}\), we get: \(a^{b \times \frac{1}{b}} = b^{a \times \frac{1}{b}}\) \(a^1 = b^{\frac{a}{b}}\) \(a = b^{\frac{a}{b}}\)
This gives us a relationship between \(a\) and \(b^{\frac{a}{b}}\).
Final Calculation
Substitute the result \(a = b^{\frac{a}{b}}\) back into our simplified expression for \(E\):
\(E = \frac{a}{ b^{ \frac{a}{b} } }\) \(E = \frac{a}{a}\) \(E = 1\)Therefore, the value of the given expression is 1.
Similar Questions
- If \(\alpha\) and \(\beta\) are the roots of the equation \(\log_{10} [998+\sqrt{x^2-18x+76}] = 3\) then what is \((\alpha - \beta)^2\) equal to?
If \(x^4 + y^4 = 14x^2 y^2\), then consider the following :
I. \(\log_{10}(x^2 + y^2)\) = \(\log_{10} x + \log_{10} y + 2\log_{10} 2\)
II. \(\log_{10}(x^2-y^2)=\log_{10} x + \log_{10} y\) +\(\log_{10}2+0.5\log_{10} 3\)
Which of the above is/are correct?
- If \(\log_{10}2 = 0.301\) and \(\log_{10} 3 = 0.477\), then what is the number of digits in the expansion of \(60^{60}\) ?
If log10(100001 - 4x)/(5 - x) = 1, then what is x equal to?
What is √17 - 4√15 + √8 - 2√15 equal to?
- If \(\log_{10}(80)=p\), \(\log_{10}(45)=q\) and \(\log_{10}(216)=r\), then what is \(\log_{10}(384)\) equal to ?
Consider the following statements in respect of common logarithms :
I. The logarithm of a number greater than 100 but less than 1000 lies between 2 and 3.
II. The logarithm of a positive number less than unity is negative.
Which of the statements given above is/are correct?
What is the number of zeros immediately after the decimal point in \((0.5)^{1000}\)? (Given that \(\log_{10}2 = 0.30103\))
A sum of money at the rate of \(5\%\) per annum compounded annually becomes \(n\) times in \(100\) years. What is the value of \(n\)? (Given \(\log_{10}2=0.301\), \(\log_{10}3=0.477\) and \(\log_{10}7=0.845\))
Important Questions from Logarithms
Given that $132^{0.14} = x$, $132^{0.26} = y$ and $x^z = y^2$, then the value of z is close to:
For \(x \ge y > 1\), let \(\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right) = k\), then the value of \(k\) can never be equal to
If \(\log_ba = p\), \(\log_dc = 2p\) and \(\log_fe = 3p\), then what is \((ace)^{\frac{1}{p}}\) equal to ?
- What is the number of solutions of \(\log_4(x-1) = \log_2(x - 3)\)?
- Let \(p = \ln(x)\), \(q = \ln(x^3)\) and \(r = \ln(x^5)\), where \(x > 1\). Which of the following statements is/are correct?
I. \(p, q\) and \(r\) are in AP.
II. \(p, q\) and \(r\) can never be in GP.
Select the answer using the code given below.
