Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{5}{\sqrt{10}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sqrt{10}}{2}\).

Step 1: Identify the square root in the denominator

The square root in the denominator is \(\sqrt{10}\).

Step 2: Rationalize the denominator

Rationalize the denominator by multiplying both the numerator and the denominator by \(\sqrt{10}\). This gives \(\frac{5*\sqrt{10}}{\sqrt{10}*\sqrt{10}}\).

Step 3: Simplify the expression

Simplify the expression to \(\frac{5*\sqrt{10}}{10}\).

Step 4: Simplify further

Simplify further by dividing both the numerator and the denominator by 5. This leads to a final answer of \(\frac{\sqrt{10}}{2}\).

Key Concepts

Simplifying Expressions

When working with algebraic expressions, simplifying means making the expression as straightforward as possible without changing its value. Think of it as tidying up the equation. You want all similar or like terms grouped together, constants reduced, and any possible factors factored out.
An expression like \( \frac{5\sqrt{10}}{10} \) can be simplified by looking for common factors in the numerator and the denominator. In this instance, both 5 and 10 can be divided by 5, so you achieve \( \frac{\sqrt{10}}{2} \).

• Reduce fractions: Always check if the numbers in the fraction have a common divisor.
• Factorize when needed: Especially around terms enclosed within square roots.

Mastering the art of simplification can be extremely valuable, as it makes handling expressions much easier!

Square Roots

Square roots often appear in mathematical expressions, and understanding how to work with them is essential. The square root of a number \( x \), written as \( \sqrt{x} \), is a value that, when multiplied by itself, gives \( x \). For instance, since \( 3 \times 3 = 9 \), we have \( \sqrt{9} = 3 \).
In algebra, square roots often need to be "rationalized" when they are found in the denominator of a fraction. This means you want to ensure that there are no square roots left in the denominator. We accomplish this by multiplying the numerator and the denominator by the square root we want to eliminate.

• Understanding multiplication of square roots: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
• Removing square roots from denominators: Use the property \( \sqrt{x} \times \sqrt{x} = x \) to get a rational number.

Algebraic Fractions

Algebraic fractions are simply fractions where the numerator, the denominator, or both, are algebraic expressions. These fractions often involve variables and can be simplified, combined, or manipulated in different ways.
When dealing with algebraic fractions, one crucial task is ensuring that the denominator doesn't contain irrational numbers, especially square roots. Rationalizing these fractions involves eliminating the square roots in the denominator, as seen in the transformation from \( \frac{5}{\sqrt{10}} \) to \( \frac{\sqrt{10}}{2} \).

• Equal treatment: Whatever operation you perform on the denominator, do the same on the numerator.
• Maintaining balance: Remember, rationalizing isn’t changing the value; it only makes the fraction easier to handle.

The ability to manipulate and simplify algebraic fractions is a foundational skill in algebra, paving the way for more complex operations later on.