Problem 40 - Algebra 1

Question

Simplify the expression. $$\frac{5}{\sqrt{7}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5\sqrt{7}}{7}\)

Step 1: Identify the expression

The given expression is \(\frac{5}{\sqrt{7}}\). The objective is to remove the square root from the denominator.

Step 2: Rationalize the denominator

To do this, multiply the numerator and the denominator by the square root of 7. That is, \(\frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}\).

Step 3: Calculate the new expression

Applying multiplication to the numerator and the denominator separately gives \(5*\sqrt{7}/7\). The square root of a number squared is the number itself.

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is fundamental in mastering algebra. Simplifying means to make an expression as simple as possible without changing its value. It usually involves combining like terms, factoring, expanding expressions, and rationalizing denominators.

For example, when you encounter an expression such as \frac{5}{\(\sqrt{7}\)}, the goal is to simplify the expression so it is easier to understand or further manipulate. We begin by identifying terms that can be combined or manipulated. In this case, there is no combining of like terms or factoring to do, but we can see that the square root in the denominator is an opportunity to simplify.

To truly simplify the expression fully, we rationalize the denominator. This process involves multiplying both the numerator and the denominator by the conjugate or same term as the denominator's radical part, which in this instance is \(\sqrt{7}\), to eliminate the square root and leave the denominator in a more simplified form.

Square Roots

The notion of square roots is a keystone in mathematics, particularly within algebra. A square root, symbolized as \(\sqrt{x}\), represents a value that, when multiplied by itself, produces the number x. For example, because 3 * 3 equals 9, we say that \(\sqrt{9}\) is 3. Square roots are the opposite operation of squaring a number.

Understanding how square roots interact with other mathematical operations is essential for simplifying algebraic expressions. When you have an expression like \(\frac{5}{\sqrt{7}}\), the presence of \(\sqrt{7}\) in the denominator prompts the need for rationalization to maintain proper mathematical form and ensure all expressions are free of radicals in the denominator.

Multiplying by \(\sqrt{7}\) effectively squares the denominator, which then simplifies to the radicand - the number under the radical symbol. This process transforms the irrationally denominated fraction into a rational expression.

Rational Expressions

Rational expressions are fractions involving polynomials in both the numerator and the denominator. Much like simplifying numerical fractions, simplifying rational expressions involves eliminating common factors and reducing the expression to its simplest form. However, when dealing with square roots, a special process called rationalizing the denominator comes into play.

To rationalize a denominator means to eliminate any radicals present. In the context of our example, \(\frac{5}{\sqrt{7}}\), multiplying the numerator and denominator by \(\sqrt{7}\) changes the denominator from an irrational square root to a rational number. After this multiplication, our expression becomes \(\frac{5\sqrt{7}}{7}\), which is a rational expression -- the square root is now only in the numerator, and the denominator is an integer.

Rational expressions are easier to work with, especially when adding, subtracting, or comparing fractions. They make it possible to perform a wider array of algebraic operations and are generally the preferred form for final answers in algebra.