"To calculate: The simplified form of the expression..." was answered by our experts

College
Calculus and Analysis
Integral Calculus
Rational exponents and radicals

Wribreeminsl

Answered question

2020-12-16

To calculate: The simplified form of the expression

\frac{\sqrt{12}}{\sqrt{x + 1}} .

Answer & Explanation

gwibdaithq

Skilled2020-12-17Added 84 answers

Formula used:
Product property of radicals: $\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{a b}$
Power property of radicals: ${(a^{m})}^{n} = a^{m n}$
Calculation:
Consider the provided expression, $\frac{\sqrt{12}}{\sqrt{x + 1}}$
Multiply numerator and denominator by $\sqrt{x + 1}$ so that the radicand in the denominator is a perfect square.
Therefore,
$\frac{\sqrt{12}}{\sqrt{x + 1}} = \frac{\sqrt{12}}{\sqrt{x + 1}} \cdot \frac{\sqrt{x + 1}}{\sqrt{x + 1}}$
Here, $x > - 1$
Apply the product property of radicals:
$\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{a b}$
Therefore,
$\frac{\sqrt{12}}{\sqrt{x + 1}} = \frac{\sqrt{4 \cdot 3} \sqrt{x + 1}}{\sqrt{(x + 1) (x + 1)}}$
$= \frac{\sqrt[2]{3} \sqrt{x + 1}}{\sqrt{{(x + 1)}^{2}}}$
Apply power property ${(a^{m})}^{n} = a^{m n} in expression \sqrt{{(x + 1)}^{2}} .$
Therefore,
$\sqrt{{(x + 1)}^{2}} = {({(x + 1)}^{2})}^{\frac{1}{2}}$
$= {(x + 1)}^{2 \cdot \frac{1}{2}}$
$= {(x + 1)}^{1}$
$= (x + 1)$
Therefore,
$\frac{\sqrt{12}}{\sqrt{x + 1}} = \frac{\sqrt[2]{3} \sqrt{x + 1}}{x + 1}$
Apply the product property of radicals: $\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{a b}$ in the numerator,
$\sqrt{3} \sqrt{x + 1} = \sqrt{3 (x + 1)}$
$= \sqrt{3 x + 3}$
Hence,
$\frac{\sqrt{12}}{\sqrt{x + 1}} = \frac{\sqrt[2]{3 x + 3}}{x + 1}$
Therefore, the simplified form of the expression $\frac{\sqrt{12}}{\sqrt{x + 1}} is \frac{\sqrt[2]{3 x + 3}}{x + 1} .$

Jeffrey Jordon

Expert2021-11-03Added 2605 answers

Answer is given below (on video)

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