Question

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\left(9-x^{2}\right)^{1 / 2}+x^{2}\left(9-x^{2}\right)^{-1 / 2}}{9-x^{2}} \quad-3

Short answer

\(\frac{9}{(9-x^{2})^{3/2}}\)

Step-by-step solution

- Simplify the numerator

The numerator is \(\left(9-x^{2}\right)^{1 / 2}+x^{2}\left(9-x^{2}\right)^{-1 / 2}\). First, rewrite the terms as radicals: \(\sqrt{9-x^{2}}+\frac{x^{2}}{\sqrt{9-x^{2}}}\).

- Find a common denominator

Combine the terms under a single denominator: \(\sqrt{9-x^{2}}=\frac{\sqrt{9-x^{2}}^2}{\sqrt{9-x^{2}}}=\frac{9-x^2}{\sqrt{9-x^{2}}}\). So, the expression becomes \(\frac{9-x^2}{\sqrt{9-x^{2}}}+\frac{x^{2}}{\sqrt{9-x^{2}}}=\frac{9-x^{2}+x^{2}}{\sqrt{9-x^{2}}}=\frac{9}{\sqrt{9-x^{2}}}\).

- Simplify the entire expression

The entire expression is now \(\frac{\frac{9}{\sqrt{9-x^{2}}}}{9-x^{2}}\). Dividing the numerator by the denominator gives: \(\frac{9}{\sqrt{9-x^{2}}\left(9-x^2\right)}=\frac{9}{(9-x^2)^{3/2}}\).

Key concepts

Rational Exponents

Rational exponents are another way to express roots. The expression \(a^{1/n}\) is the same as \(\root n\text{ }a\). This concept simplifies complex expressions and can make them easier to manipulate, especially when combining terms or integrating expressions. When you see \(a^{m/n}\), it means \((\root n\text { }a)^m\). For example, \(\root3\text{ }8 = 8^{1/3}\), which simplifies to 2 because 2^3 = 8. Understanding rational exponents helps in manipulating algebraic expressions more flexibly and efficiently.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. The symbol \(\root\text{ }\text{}\) indicates a root, and the number inside the root symbol is called the radicand. For example, \(\root2\text{ }9\) is the square root of 9, which equals 3. There are rules for working with radicals that make problems easier to solve:

• To simplify radicals, factor the radicand into its prime factors.
• Combine like terms under the same radical.
• Rationalize the denominator by eliminating any roots in the denominator of a fraction.

These steps are essential for effectively simplifying and solving radical expressions.

Combining Terms

Combining terms involves simplifying expressions by adding or subtracting like terms. Like terms have the same variable raised to the same power. For example, in \(\root\text{ }\text{ }a \text{ + } b \text{ -c}\), the terms with the same radical are combined. To combine these like terms, follow these steps:

• First, simplify each term individually.
• Then, find a common denominator if the terms are fractions.
• Combine the numerators over the common denominator.

By mastering combining terms, you can simplify complex expressions and solve equations effectively.