Problem 339 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{5+\sqrt{125}}{15} $$

Step-by-Step Solution

Answer

\(\frac{1 + \sqrt{5}}{3}\)

Step 1: Simplify the Square Root

First, simplify the square root in the numerator. Since \(\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}\), the expression inside the numerator can be rewritten.

Step 2: Rewrite the Numerator

Rewrite the numerator using the simplified square root: \(5 + \sqrt{125} = 5 + 5\sqrt{5}\). This allows combining similar terms.

Step 3: Factor the Numerator

Factor out the common factor of 5 in the numerator: \(5 + 5\sqrt{5} = 5(1 + \sqrt{5})\).

Step 4: Simplify the Fraction

The expression now is \(\frac{5(1 + \sqrt{5})}{15}\). Simplify the fraction by dividing both the numerator and the denominator by 5: \(\frac{5(1 + \sqrt{5})}{15} = \frac{1 + \sqrt{5}}{3}\).

Step 5: State the Final Simplified Form

The simplified form of the original expression is \(\frac{1 + \sqrt{5}}{3}\).

Key Concepts

square roots

Square roots are fundamental in many areas of mathematics. A square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). For example, \(\sqrt{25} = 5\) because \(5^2 = 25\). In our exercise, we have the term \(\sqrt{125}\). We simplify this by recognizing it can be factored into a product of square roots: \(\sqrt{125} = \sqrt{25\cdot5} = \sqrt{25}\cdot\sqrt{5} = 5\sqrt{5}\). Breaking the term into factorable components often makes the simplification more straightforward.

factoring

Factoring involves expressing an expression as a product of its factors. In our exercise, after simplifying \(\sqrt{125}\) to \(5\sqrt{5}\), we rewrite the numerator: \(5 + 5\sqrt{5}\). Here, you can see that 5 is a common factor in both terms. So, we factor out the 5: \(5 + 5\sqrt{5} = 5(1 + \sqrt{5})\). Factoring makes it easier to manage and simplify algebraic expressions, especially within fractions.

simplification

Simplification refers to reducing an expression to its simplest form. Starting with \(\frac{5 + \sqrt{125}}{15}\), we first simplify the square root: \(\sqrt{125} = 5\sqrt{5}\). We rewrite the expression as \(\frac{5 + 5\sqrt{5}}{15}\). Factoring the numerator, we get \(\frac{5(1 + \sqrt{5})}{15}\). Finally, we simplify the fraction by dividing both numerator and denominator by 5, leading us to the simplified expression \(\frac{1 + \sqrt{5}}{3}\). Each step simplifies the expression further until we reach the most reduced form.

fractions

A fraction represents a division of one quantity by another. In the problem \(\frac{5 + \sqrt{125}}{15}\), the goal is to simplify this fraction. Simplifying a fraction often involves both factoring and reducing. In our steps, after breaking down the terms in the numerator and rewriting as a factorable expression, we achieve \(\frac{5(1 + \sqrt{5})}{15}\). We then recognize that both the numerator and denominator are divisible by 5. Simplifying the division, we find the final solution \(\frac{1 + \sqrt{5}}{3}\). Always look for common factors to simplify fractions efficiently.