Simplify-x-4-x-4-x-4-x-4-x-8-4-x-2-12x-32-

Question Number 198814 by hardmath last updated on 24/Oct/23

Simplify :

\frac{(\sqrt{x + 4 \sqrt{x - 4}} + \sqrt{x - 4 \sqrt{x - 4}}) \sqrt{x - 8}}{4 \sqrt{x^{2} - 12 x + 32}}

Answered by a.lgnaoui last updated on 24/Oct/23

= \frac{1}{4} \sqrt{\frac{x + 4 \sqrt{x - 4}}{x - 4}} + \frac{1}{4} \sqrt{\frac{x - \sqrt{x - 4}}{x - 4}}

= \frac{\sqrt{x + 4 \sqrt{x - 4}} + \sqrt{x - 4 \sqrt{x - 4}}}{4 \sqrt{x - 4}}

= \frac{\sqrt{(x - 4) + 4 \sqrt{x - 4} + 4}}{4 \sqrt{x - 4}} + \frac{\sqrt{x - 4) - 4 \sqrt{x - 4} + 4}}{4 \sqrt{x - 4}}

\frac{\sqrt{{(\sqrt{x - 4} + 2)}^{2}}}{4 \sqrt{x - 4}} + \frac{\sqrt{{(\sqrt{x - 4}) - 2)}^{2}}}{4 \sqrt{x - 4}}

= \frac{\sqrt{x - 4} + 2}{4 \sqrt{x - 4}} + \frac{\sqrt{x - 4} - 2}{4 \sqrt{x - 4}} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

Commented by hardmath last updated on 28/Oct/23

thanks professor

Answered by Rasheed.Sindhi last updated on 25/Oct/23

\frac{(\sqrt{x + 4 \sqrt{x - 4}} + \sqrt{x - 4 \sqrt{x - 4}}) \sqrt{x - 8}}{4 \sqrt{x^{2} - 12 x + 32}}

\frac{(\sqrt{x + 4 \sqrt{x - 4}} + \sqrt{x - 4 \sqrt{x - 4}}) \sqrt{x - 8}}{4 \sqrt{(x - 4) (x - 8)}}

x > 8 \Rightarrow x - 4 > 4 \Rightarrow \sqrt{x - 4} > 2

\frac{\sqrt{x + 4 \sqrt{x - 4}} + \sqrt{x - 4 \sqrt{x - 4}}}{4 \sqrt{x - 4}}

L e t \sqrt{x - 4} = y

x > 8 \Rightarrow y = \sqrt{x - 4} > 2

x = y^{2} + 4

\frac{\sqrt{y^{2} + 4 + 4 y} + \sqrt{y^{2} + 4 - 4 y}}{4 y}

\frac{∣ y + 2 ∣ + ∣ y - 2 ∣}{4 y}

∵ y > 2

∴ \frac{∣ y + 2 ∣ - ∣ y - 2 ∣}{4 y} = \frac{(y + 2) + (y - 2)}{4 y} = \frac{2 y}{4 y} = \frac{1}{2}

Commented by hardmath last updated on 28/Oct/23

thanks professor