Problem 28 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \sqrt{75}+2 \sqrt{48}-5 \sqrt{3} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(8 \sqrt{3}\).

Step 1: Breaking Down the Radicals

Start with expressing numbers under each radical as the product of a perfect square and another number. So, we get: \[ \sqrt{75} = \sqrt{25*3},\ 2 \sqrt{48} = 2 \sqrt{16*3}, \ -5 \sqrt{3} = -5 \sqrt{3} \]

Step 2: Simplify the Radicals

Use the property that the square root of the product of two numbers is the product of the square roots of those numbers, to simplify: \[ \sqrt{25*3} = \sqrt{25} * \sqrt{3} = 5 \sqrt{3},\ 2 \sqrt{16*3} = 2(\sqrt{16} * \sqrt{3}) = 2*4 \sqrt{3} = 8 \sqrt{3}, \ -5 \sqrt{3} = -5 \sqrt{3} \] This simplifies the initial expression to \( 5 \sqrt{3}+8 \sqrt{3}-5 \sqrt{3} \).

Step 3: Combine Like Terms

Combine the like terms to get the final simplified result. We have: \[5 \sqrt{3}+8 \sqrt{3}-5 \sqrt{3} = 8 \sqrt{3} \]

Key Concepts

Understanding Radicals

A radical expression is simply an expression that includes a square root, cube root, or other root. The most common radical is the square root, which is represented by the symbol \( \sqrt{} \). Radicals can look complicated, but they follow specific rules that make them simpler to handle.

• Every radical can be expressed in the form of \( \sqrt{a} \), where \( a \) is a number or an algebraic expression.
• Radicals can also be split into two numbers under the square root sign, like \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).

When simplifying radicals, always look to express the number inside the radical as a product of a perfect square (a number that has an integer square root) and another number. This helps to "extract" the square root of the perfect square, simplifying the expression.

Combining Like Terms

In mathematics, combining like terms is a fundamental skill, especially in algebra. Like terms are terms in a mathematical expression that have the same variable raised to the same power. For terms containing radicals, the radical part must also be identical.

• In the expression \( a \sqrt{b} + c \sqrt{b} \), \( a \) and \( c \) are the coefficients that can be combined.
• To combine like terms, simply add or subtract the coefficients.

In our example, \( 5 \sqrt{3} + 8 \sqrt{3} - 5 \sqrt{3} \) contains like terms because they all involve \( \sqrt{3} \). You combine them by simply operating on the numbers outside the square roots, which gives \( 8 \sqrt{3} \).

The Role of Perfect Squares

A perfect square is an important concept to grasp when working with radicals. Perfect squares are numbers whose square roots are integers. For example, 1, 4, 9, 16, 25, etc., are perfect squares.

• If you have a perfect square inside a square root, taking the square root is straightforward, e.g., \( \sqrt{25} = 5 \).
• Identifying perfect square factors in a radical like \( \sqrt{75} \) (which is \( \sqrt{25 \times 3} \)) allows you to simplify the expression to \( 5 \sqrt{3} \).

Finding and using perfect squares helps in breaking down complex radicals into simpler terms, making mathematical expressions easier to work with.

Exploring Mathematical Expressions

Mathematical expressions encompass numbers, variables, operators, and sometimes radicals. Understanding how to manipulate these expressions is key to solving mathematical problems.

• Expressions can include various types of terms like constants (such as \( 5 \)), coefficients (such as the \( 2 \) in \( 2 \sqrt{3} \)), and radicals.
• To simplify, always look for opportunities to combine like terms, factor, or apply mathematical properties like that of radicals.

By practicing how to work with these elements, such as in the simplification of radicals and combining like terms, better mathematical fluency is developed, allowing for even the most complex expressions to be tackled with confidence.