16.if 2^(x)=3^(y)=12^(z) Show That (1)/(z)=(1)/(y)+(2)/(x)

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Jul 01, 2024

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If 2^(x) = 3^(y) = 12^(z), Show that (1)/(z) = (1)/(y) + (2)/(x)

In this problem, we are given three equations: 2^(x) = 3^(y) = 12^(z). We need to prove that (1)/(z) = (1)/(y) + (2)/(x).

Step 1: Express 12 in terms of 2 and 3

First, let's express 12 in terms of 2 and 3:

12 = 2^2 × 3

Now, substitute this expression into the given equation:

2^(x) = 3^(y) = (2^2 × 3)^(z)

Step 2: Simplify the equation

Using the property of exponentiation, we can simplify the equation:

2^(x) = 3^(y) = 2^(2z) × 3^(z)

Step 3: Equate the exponents

Since the bases are the same (2), we can equate the exponents:

x = 2z ... (1) y = z ... (2)

Step 4: Substitute the values into the required equation

Now, substitute the values of x and y in terms of z into the required equation:

(1)/(z) = (1)/(y) + (2)/(x) (1)/(z) = (1)/(z) + (2)/(2z)

Step 5: Simplify the equation

Simplify the equation:

(1)/(z) = (1)/(z) + (1)/(z)

Since the equation is true, we have proved that:

(1)/(z) = (1)/(y) + (2)/(x)

Therefore, the given statement is true.