# If $X^{y^z}=1, Y^{z^x}=125$ and $Z^{y^x}=243$ (x, y and z are natural numbers), then what is the value of 9x + 10y - 18z?

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If $X^{y^z}=1, Y^{z^x}=125$ and $Z^{y^x}=243$ (x, y and z are natural numbers), then what is the value of 9x + 10y - 18z?
1. 18
2. 15
3. 12
4. 5

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Correct Answer - Option 4 : 5

$x^{(y^{z} )}=1$ …… (i)

∴ x = 1, because for any value of yz

If x = 1, then, $x^{y^{z} }$ remain 1.

∴ x = 1

$y^{z^{x }}=125$…… (ii)

We can verify value from this equation by putting x = 1, y = 5 and z = 3

$5^{3^{1}}=125$

Again,

$z^{y^{x}}=243$ …… (iii)

$z^{y^{x}}=3^{5^{1}}=243$

∴ It is clear that –

⇒ x = 1, y = 5 and z = 3

The value of 9x + 10y – 18z

⇒ 9 + 50 – 54

⇒ 5