Question

$if \: 3 {}^{x} = 5 {}^{y} = (75) {}^{z} \: show \: that \: z = \frac{xy}{2x + y}$
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Answer 1

Answer:

refer to the pic

Step-by-step explanation:

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Answer 2

$<font color="purple">$

★FIND:

➡️$\bold{z = \frac{xy}{2x + y}}$

★GIVEN,

➡️$\bold{3 {}^{x} = 5 {}^{y} = (75) {}^{z}}$

★SOLUTION:

➡️$\bold{3 {}^{x} = 5 {}^{y} = (75) {}^{z}}$

➡️$\bold{3^x=5^y=75^z=k}$ says

➡️$\bold{3^x=k,\:5^y=k,\:75^z=k}$

➡️$\bold{3 = k {}^{1/x}, \: 5 = k {}^{1/ \: y}, \: 75 = k {}^{1/z} }$

★NOW,

➡️$\bold{75=3×5×5}$

➡️$\bold{k {}^{1/z} = k {}^{1/x} \times k {}^{1/y} \times k {}^{1/y}}$

➡️$\bold{k {}^{1/z} = k {}^{1/x + 2/y}}$$\bold{(since \: a {}^{m} a {}^{n} = a {}^{m + n})}$

➡️$\bold{1/z = 1/x + 2/y}$

➡️$\bold{1 /z= y + 2x/xy}$

➡️$\bold{z = \frac{xy}{2x + y}}$