If 5^x = 100^y = 20^z for non-zero real numbers x, y, and z, which of the following expres… — GMAT

If $5^x = 100^y = 20^z$ for non-zero real numbers $x$ , $y$ , and $z$ , which of the following expressions is equal to $z$ ?

\frac{xy}{x - y}

By setting

5^x = 100^y = 20^z = K

, we rewrite each base as a power of

K

5 = K^{1/x}

100 = K^{1/y}

, and

20 = K^{1/z}

. Using the numerical relationship

20 = \frac{100}{5}

, we substitute to obtain

K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

. Equating exponents gives

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}

. Taking the reciprocal yields the expression stating that

z

equals

\frac{xy}{x - y}

Set the common exponential expressions equal to a constant

K

Let

5^x = 100^y = 20^z = K

, where

K > 0

and

K \neq 1

Equating exponential terms to a shared parameter allows conversion from variable powers to fractional exponents.

Express the base numbers

5

100

, and

20

in terms of

K

5 = K^{1/x}

100 = K^{1/y}

, and

20 = K^{1/z}

Taking the

x

-th,

y

-th, and

z

-th roots of both sides isolates each base number.

Relate the base numbers using arithmetic division.

20 = \frac{100}{5} \implies K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

Dividing powers with the same base subtracts their exponents.

Equate the exponents and solve for

z

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy} \implies z = \frac{xy}{x - y}

Taking common denominators and inverting the fraction yields

z

in terms of

x

and

y

Relating different bases with variable exponents by setting them equal to a common constant and converting to reciprocal exponents.

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