Problem 58 - Algebra 1

Question

Which is the simplified form of \(4 \frac{\sqrt{125}}{\sqrt{25}} ?\) (A) \(2 \sqrt{5}\) (B) \(4 \sqrt{5}\) (C) \(20 \sqrt{5}\) (D) \(\frac{4 \sqrt{5}}{5}\)

Step-by-Step Solution

Answer

(B) \(4 \sqrt{5}\)

Step 1: Simplify the Square Roots

Start by simplifying each square root: \(\sqrt{125} = \sqrt{25} \times \sqrt{5} = 5 \sqrt{5}\) \(\sqrt{25} = 5\)

Step 2: Substitute the Simplified Square Roots

Now, we'll substitute the simplified square roots into our original expression:\(4 \frac{\sqrt{125}}{\sqrt{25}} = 4 \frac{5 \sqrt{5}}{5}\)

Step 3: Cancel Out Common Factors

Next, you can cancel out the common factor of 5 in the numerator and denominator which results in:\(4 \frac{5 \sqrt{5}}{5} = 4 \sqrt{5}\)

Key Concepts

Rational Expressions

A rational expression is simply a fraction where the numerator and the denominator are both polynomials. These can also include expressions with square roots in the numerator or denominator. Simplifying rational expressions is key to making them easier to work with in mathematical computations. When you encounter square roots within these expressions, the goal is to simplify them as much as possible to get to the root (no pun intended!) of the problem.
In the example given, you see a rational expression involving square roots: \[ 4 \frac{\sqrt{125}}{\sqrt{25}} \]The numerator \( \sqrt{125} \) and the denominator \( \sqrt{25} \) are both square roots. To simplify, focus on reducing both the numerator and the denominator to their simplest terms.
This simplifies our calculations and helps in avoiding errors when working with larger, more complex rational expressions.

Square Root Properties

Square roots have special properties that can be used to simplify expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For simplification, a key property is: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] where \( a \) and \( b \) are positive numbers. This property allows us to break down complex square roots into more manageable pieces:
You can easily simplify square roots by looking for perfect square factors. In our example, the square root of 125 can be broken down as follows:

• \( \sqrt{125} = \sqrt{25 \times 5} \) because 25 is a perfect square.
• This leads to \( \sqrt{125} = \sqrt{25} \times \sqrt{5} = 5 \sqrt{5} \).

Additionally, \( \sqrt{25} = 5 \), since 25 is a perfect square. These simplifications provide us with a simpler form to insert back into the rational expression.

Algebraic Simplification

Algebraic simplification is essentially about making an expression as simple as possible by applying mathematical properties. Involving steps like reducing terms, canceling out common factors, and other algebraic rules, this process makes calculations more straightforward.
In the original expression:\[ 4 \frac{5 \sqrt{5}}{5} \]we see that the fraction \( \frac{5 \sqrt{5}}{5} \) contains the number 5 in both the numerator and the denominator. By canceling out the common factor of 5, we simplify the expression:

• \( \frac{5}{5} = 1 \), so \( 5 \sqrt{5} \rightarrow \sqrt{5} \).
• This leads to \( 4 \cdot \sqrt{5} = 4 \sqrt{5} \).

Through this algebraic simplification, we can confidently determine the simplified form. Being able to easily simplify expressions is crucial in algebra, helping to eliminate complexity and highlight the underlying mathematical structure.