Question

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\sqrt{1+x}-x \cdot \frac{1}{2 \sqrt{1+x}}}{1+x} \quad x>-1$$

Short answer

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

Step-by-step solution

- Rewrite the numerator

Rewrite the numerator to simplify it. The numerator is \(\sqrt{1+x}-x \cdot \frac{1}{2 \sqrt{1+x}}\). This can be expressed as \( \sqrt{1+x} - \frac{x}{2 \sqrt{1+x}} \).

- Combine fractions in the numerator

To combine the fractions in the numerator, find a common denominator. The common denominator is \(2\sqrt{1+x} \). Rewrite each term with this common denominator: \[ \frac{2(1+x)}{2\sqrt{1+x}} - \frac{x}{2\sqrt{1+x}} \].

- Simplify the numerator

Combine the terms in the numerator: \[ \frac{2(1+x) - x}{2\sqrt{1+x}} = \frac{2 + 2x - x}{2\sqrt{1+x}} = \frac{2 + x}{2\sqrt{1+x}} \].

- Write the entire expression as a single quotient

Now, divide the simplified numerator by the given denominator \(1+x \): \[ \frac{\frac{2 + x}{2\sqrt{1+x}}}{1+x} \].

- Simplify the final quotient

To divide by a fraction, multiply by its reciprocal: \[ \frac{2 + x}{2\sqrt{1+x}} \times \frac{1}{1+x} = \frac{2 + x}{2\sqrt{1+x}(1+x)} \].

- Final result

Therefore, the simplified single quotient is: \[ \frac{2 + x}{2(1+x)\sqrt{1+x}} \].

Key concepts

Positive Exponents

An exponent tells us how many times to multiply a number by itself. Positive exponents are straightforward and easier to manage. For example, in the exercise, the term \(\frac{1}{2\sqrt{1+x}}\) is part of the numerator. To simplify expressions involving exponents, it's important always to seek a form using positive exponents. Here are some rules to remember for positive exponents:

• When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Any number raised to the power of 0 is 1: \(a^0 = 1\).

Working with positive exponents ensures easier manipulation and simplification of algebraic expressions, setting a solid base for more advanced mathematics.

Common Denominators

When combining fractions, having a common denominator is crucial. It allows us to add or subtract fractions easily. For instance, in the exercise, to combine \(\sqrt{1+x}\) and \(\frac{x}{2\sqrt{1+x}}\), we find the common denominator, which is \(2\sqrt{1+x}\). This step is essential because:

• A common denominator aligns the fractions, making addition or subtraction straightforward.
• It simplifies expressions and prepares them for further operations.

Here’s a quick guide to finding a common denominator:

• Identify the denominators of the fractions involved.
• Find the least common multiple (LCM) of these denominators.
• Rewrite each fraction with the LCM as the new denominator.

Finding common denominators simplifies the process, allowing us to focus on the numerator.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This concept is particularly important in algebra to make expressions easier to handle. In the step-by-step solution above, simplifying fractions helps write the expression as a single quotient. Here’s how to simplify fractions effectively:

• Factor both the numerator and the denominator.
• Cancel out common factors from the numerator and the denominator.
• Recombine the simplified terms.

For example, in the final step of the exercise, we simplify \(\frac{2 + x}{2(1 + x)\sqrt{1 + x}}\) by recognizing that no further simplification is possible without violating the requirement to maintain positive exponents and radicals. Simplifying is about making expressions cleaner and easier to interpret, vital for progressing in algebra and calculus.