√(e^4x e^2x) =√((e^2x)^2 (e^x)^2) =√((e^2x e^x)(e^2x+e^x)) =√(e^2x e^x)√(e^2x+e^x) ∴√(e^4x e^2x) = √(e^2x e^x) √(e^2x+e^x) ...

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Byju's Answer

Standard VIII

Mathematics

Finding Square of a Number Using Identity

Solve: √e4x...

Question

Solve: $\sqrt{e^{4 x} - e^{2 x}}$ .

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Solution

$\sqrt{e^{4 x} - e^{2 x}}$
$= \sqrt{(e^{2 x})^{2} - (e^{x})^{2}}$
$= \sqrt{(e^{2 x} - e^{x}) (e^{2 x} + e^{x})}$
$= \sqrt{e^{2 x} - e^{x}} \sqrt{e^{2 x} + e^{x}}$
$∴ \sqrt{e^{4 x} - e^{2 x}}$ = $\sqrt{e^{2 x} - e^{x}}$ $\sqrt{e^{2 x + e^{x}}}$

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{lim}_{x \to 0} \frac{e^{3 x} - e^{2 x}}{e^{4 x} - e^{3 x}}

Let I $= \int \frac{e^{x}}{e^{4 x} + e^{2 x} + 1} d x . J = \int \frac{e^{- x}}{e^{- 4 x} + e^{- 2 x} + 1} d x,$ Then, for an arbitrary constant c, the value of J-I equals

Q. If

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Solve: √e4x−e2x.

√e4x−e2x=√(e2x)2−(ex)2=√(e2x−ex)(e2x+ex)=√e2x−ex√e2x+ex∴√e4x−e2x= √e2x−ex√e2x+ex

Solve: $\sqrt{e^{4 x} - e^{2 x}}$ .

$\sqrt{e^{4 x} - e^{2 x}}$
$= \sqrt{(e^{2 x})^{2} - (e^{x})^{2}}$
$= \sqrt{(e^{2 x} - e^{x}) (e^{2 x} + e^{x})}$
$= \sqrt{e^{2 x} - e^{x}} \sqrt{e^{2 x} + e^{x}}$
$∴ \sqrt{e^{4 x} - e^{2 x}}$ = $\sqrt{e^{2 x} - e^{x}}$ $\sqrt{e^{2 x + e^{x}}}$