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Comparison of Quantities Using Exponents

If x1/p = y...

Question

If $x^{1 / p} = y^{1 / q} = z^{1 / r}$ and $x y z = 1,$ then the value of $p + q + r = ?$

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Solution

$x^{1 / p} = y^{1 / q} = z^{1 / r} = k$ (say)
$\Rightarrow x = k^{p}, y = k^{q}, z = k^{r}$
$x y z = 1$ (given)
$k^{p} . k^{q} . k^{r} = 1$
$k^{(p + q + r)} = 1$ .
$\Rightarrow (p + q + r) = 0$ .

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= ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{(a + x)} & \frac{1}{(b + x)} & \frac{1}{(c + x)} \frac{1}{(a + y)} & \frac{1}{(b + y)} & \frac{1}{(c + y)} \frac{1}{(a + z)} & \frac{1}{(b + z)} & \frac{1}{(c + z)} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ = \frac{P}{Q}

, where Q is the product of denominators. Find P

What is P - (Q + R)?

If, P = $\frac{1}{2}$ xyz, Q = $\frac{3}{2}$ xyz and R = -2 xyz

p = r (4 - r)

q = p (4 - p)

r = q (4 - q)

p \neq q \neq r \neq 0

Then find minimum value of

(p + q + r)

s = p + q + r

, then value of

∣ ∣ ∣ ∣ \begin{matrix} s + r & p & q r & s + p & q r & p & s + q \end{matrix} ∣ ∣ ∣ ∣

k = p + q + r

, then the value of

∣ ∣ ∣ ∣ \begin{matrix} k + r & p & q r & k + p & q r & p & k + q \end{matrix} ∣ ∣ ∣ ∣

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