Given `10^x = 9` , `10^y = 1/4` and `10^z = 5` evaluate `10^(((x+y)/2) - 3z)` Show your working

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We wish to evaluate the expression

`10^(((x+y)/2) - 3z)`

First, simplify the power into seperate terms involving each of `x`, `y` and `z` only:

`10^(((x+y)/2) - 3z) = 10^(x/2 + y/2 - 3z)`

Then remember that when we multiply` ` `x^a`, `x^b` and `x^c` together, the powers add.

This then gives

`10^(x/2+y/2-3z) = 10^(x/2)10^(y/2)10^(-3z)`

Finally remember that when we raise `x^a` to the power `b` , ie evaluate `(x^a)^b`, the powers are multiplied.

Hence

`10^(x/2)10^(y/2)10^(-3z) = (10^x)^(1/2)(10^y)^(1/2)(10^z)^(-3)`

Since we know that `10^x = 9` , `10^y = 1/4`  and `10^z = 5` we have that

`(10^x)^(1/2)(10^y)^(1/2)(10^z)^(-3) = sqrt(9) times sqrt(1/4) times 1/5^3`

`= 3 times 1/2 times 1/125 = 3/250 = 0.012`

Therefore

`10^(((x+y)/2) - 3z) = 0.012`

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