Problem 181 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{27}-\sqrt{75} $$

Step-by-Step Solution

Answer

The simplified expression is \( -2\sqrt{3} \).

Step 1 - Express each radicand as a product of a square number

First, find the prime factorization of each number under the square roots. \( 27 = 3 \times 3 \times 3 = 3^3 \)\( 75 = 3 \times 5 \times 5 = 3 \times 5^2 \)

Step 2 - Simplify each square root expression

Rewrite the square roots using the factored forms: \( \sqrt{27} = \sqrt{3^3} = \sqrt{3^2 \times 3} = 3\sqrt{3} \)\( \sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3} \)

Step 3 - Subtract the simplified square roots

Since both terms now have \(\sqrt{3}\), combine them:\(3\sqrt{3} - 5\sqrt{3} = (3 - 5)\sqrt{3} = -2\sqrt{3}\)

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest prime factors. Prime numbers are those that only have two divisors: 1 and themselves. To find the prime factorization of a number like 27, you repeatedly divide by the smallest prime number until you reach 1. For 27:
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
This can be expressed as: \(27 = 3 \times 3 \times 3 = 3^3 \).
For another example, take 75:
- 75 ÷ 3 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1 \(75 = 3 \times 5 \times 5 = 3 \times 5^2 \). Prime factorization simplifies understanding and addressing more complex mathematical problems.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3^2 = 9 \). When simplifying square roots, the goal is to transform the expression under the root into a product of perfect squares and other factors.
In our example: \( \sqrt{27} = \sqrt{3^3} = \sqrt{3^2 \times 3} = 3\sqrt{3} \)
\( \sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3} \).
When dealing with square roots, look for perfect squares to simplify the process.

Algebraic Expressions

Algebraic expressions include numbers, variables, and operations (+, -, *, /). They represent mathematical relationships. Learning how to manipulate and simplify these expressions is crucial for solving algebraic problems.
In this exercise, we simplified \(\sqrt{27} - \sqrt{75} \) to a form where both terms have a common radical \(\sqrt{3} \).
This is now an algebraic expression we can work with, \(3\sqrt{3} - 5\sqrt{3} \).
The key step is combining like terms: \((3 - 5)\sqrt{3} = -2\sqrt{3} \). Simplification often requires recognizing and grouping similar components.

Radicals

Radicals involve roots, such as square roots. Simplifying radicals makes expressions easier to work with and understand. In our example: \(\sqrt{27} = 3\sqrt{3} \) and \(\sqrt{75} = 5\sqrt{3} \).
Combining these gives \(3\sqrt{3} - 5\sqrt{3} = (3 - 5)\sqrt{3} = -2\sqrt{3} \).
Working with radicals often involves identifying hidden factors, pulling out perfect squares, and combining like terms. Simplifying radicals can help in solving varied mathematical problems efficiently.