Question

(x√x)^x = x^(x√x) find x​

Answer 1

SOLUTION:

Given

(x√x)ˣ = xˣ√ˣ

Method 1

Applying logarithms on both sides

log(x√x)ˣ = log xˣ√ˣ

→ x log x√x = x√x log x [ °.° log aⁿ = n log a]

→ x log x(x)¹/² = x(x)¹/² log x. [°.° √ = ^½]

→ x log x³/² = x³/² log x [ °.° a^m × aⁿ = a^(m+n)]

→ 3/2 x log x = x³/² log x

log x get cancelled

→ 3/2 x = x³/²

→ 3/2 = (x³/²)/x

→ 3/2 = x³/² × x-¹

→ 3/2 = x³/²-1

→ 3/2 = x¹/²

→ x = (3/2)²

→ x = 9/4

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Method 2:

(x√x)ˣ = x^(x√x)

→ (x.x¹/²)ˣ = x^(x.x¹/²)

→ (x³/²)ˣ = x^(x³/²)

→ 3/2 x = x³/² [ °.° a^m = aⁿ → m = n ]

→ 3/2 = x-¹ × x³/²

→ 3/2 = x¹/²

→ x = (3/2)²

→ x = 9/4

_______________________________

Hence , x = 9/4

Answer 2

${\huge{\underline{\boxed{\red{\mathcal{Answer}}}}}}$

Given:

• $\left(x\sqrt{x}\right)^x = x^{x\sqrt{x}}$

To Find:

• Value of x

Solution:

$\implies \left(x\sqrt{x}\right)^x = x^{x\sqrt{x}} \\\\\implies \left(\sqrt{x^2*x}\right)^x = x^{x\sqrt{x}} \\\\\implies \left(\sqrt{x^3}\right)^x = x^{x\sqrt{x}}} \\\\\implies x^{\frac{3}{2}*x} = x^{x\sqrt{x}}} \\\\\text{As, bases are same, we compare the powers,} \\\\\implies \dfrac{3x}{2} = x\sqrt{x} \\\\\implies \dfrac{9x^2}{4} = x^2*x \:\:\:(On\: squaring\:both\:sides) \\\\\implies 9x^2 = 4x^3 \\\\\implies 9 = 4x \:\:\:\:\:(x^2\:gets \:cut)\\\\\bold{\implies x = \dfrac{9}{4}}$

Hope it helps!!