If $(5.55)^x= (0.555)^y = 1000$, then the value of $\frac{1}{x} - \frac{1}{y}$ is

We have: $(5.55)^x = (0.555)^y = 1000$

This means we have two separate equations:

$(5.55)^x = 1000$

$(0.555)^y = 1000$

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When we have an equation like $a^x = b$, we can rewrite it as $a = b^{1/x}$. This form will help us eliminate the common value 1000.

From $(5.55)^x = 1000$:

$5.55 = 1000^{1/x}$ ...(equation 1)

From $(0.555)^y = 1000$:

$0.555 = 1000^{1/y}$ ...(equation 2)

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Since both equations equal powers of 1000, dividing them will help us find a relationship between $\frac{1}{x}$ and $\frac{1}{y}$.

Dividing equation 1 by equation 2:

$\frac{5.55}{0.555} = \frac{1000^{1/x}}{1000^{1/y}}$

Left side: $\frac{5.55}{0.555} = \frac{555}{55.5} = 10$

Right side: Using the law of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{1000^{1/x}}{1000^{1/y}} = 1000^{(1/x - 1/y)}$

So we have:

$10 = 1000^{(1/x - 1/y)}$

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To compare exponents, we need the same base on both sides.

Notice that:

$10 = 10^1$

$1000 = 10^3$

So our equation becomes:

$10^1 = (10^3)^{(1/x - 1/y)}$

Using the power rule $(a^m)^n = a^{mn}$:

$10^1 = 10^{3(1/x - 1/y)}$

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When $a^m = a^n$ (where $a > 0$ and $a

eq 1$), then$m = n$.

Since the bases are equal, the exponents must be equal:

$1 = 3\left(\frac{1}{x} - \frac{1}{y}\right)$

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Dividing both sides by 3:

$\frac{1}{x} - \frac{1}{y} = \frac{1}{3}$

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When we have two exponential equations with the same result, we can often divide them to create a relationship between the exponents. This technique is particularly useful when dealing with powers of 10, as it allows us to simplify complex exponential relationships.

Answer: $\frac{1}{3}$