Key to "The following are expressions that occur in calculus...."

High School
Algebra
Algebra I
Exponents and radicals

Wierzycaz

Answered question

2021-09-22

The following are expressions that occur in calculus. Write each expression as a single quotient in which only positive exponents and/or radicals appear.

2 \times {(x^{2} + 1)}^{\frac{1}{2}} + x^{2} \cdot \frac{1}{2} {(x^{2} + 1)}^{- \frac{1}{2}} \cdot 2 x

Answer & Explanation

delilnaT

Skilled2021-09-23Added 94 answers

Given expression:

2 \times {(x^{2} + 1)}^{\frac{1}{2}} + x^{2} \cdot \frac{1}{2} {(x^{2} + 1)}^{- \frac{1}{2}} \cdot 2 x

by simplifying given expression,

2 \times {(x^{2} + 1)}^{\frac{1}{2}} + x^{2} \times \frac{1}{2} {(x^{2} + 1)}^{\frac{- 1}{2}} \times 2 x = 2 {(x^{2} + 1)}^{\frac{1}{2}} + x^{3} {(x^{2} + 1)}^{\frac{- 1}{2}}

By using exponent rule

\frac{t}{x^{n}} = x^{- n}

2 {(x^{2} + 1)}^{\frac{1}{2}} + x^{3} {(x^{2} + 1)}^{\frac{- 1}{2}} = 2 {(x^{2} + 1)}^{\frac{1}{2}} + \frac{x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}}

To take the addition of both fraction make the same denominator by multiplying and dividing first fraction by

{(x^{2} + 1)}^{\frac{1}{2}}

2 {(x^{2} + 1)}^{\frac{1}{2}} + \frac{x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}} = \frac{2 {(x^{2} + 1)}^{\frac{1}{2}} \times {(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}} + \frac{x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}}

= \frac{2 {(x^{2} + 1)}^{\frac{1}{2}} \times {(x^{2} + 1)}^{\frac{1}{2}} + x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}}

By simplifying numerator,

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