Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ -\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}\right)\right] $$

Short answer

The simplified expression is \(-\frac{1}{9}\).

Step-by-step solution

Simplify the Innermost Parentheses

First, calculate the expression inside the innermost parentheses: \( \frac{1}{3} - \frac{1}{5} = \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \). So, the expression becomes \(-\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{2}{15}\right)\right]\).

Distribute the Fraction in Parentheses

Next, distribute \( -\frac{1}{2} \) across the result from the innermost parentheses: \(-\frac{1}{2} \times \frac{2}{15} = -\frac{2}{30} = -\frac{1}{15}\). Thus, the expression is now \(-\frac{1}{3}\left[\frac{2}{5} - \frac{1}{15}\right]\).

Simplify Expression in Brackets

Calculate the expression inside the brackets: \( \frac{2}{5} - \frac{1}{15} = \frac{6}{15} - \frac{1}{15} = \frac{5}{15} = \frac{1}{3} \). The expression becomes \(-\frac{1}{3}\left(\frac{1}{3}\right)\).

Multiply to Remove Remaining Parentheses

Finally, multiply to remove the remaining parentheses: \(-\frac{1}{3} \times \frac{1}{3} = -\frac{1}{9} \).

Final Simplified Expression

The simplified expression is \(-\frac{1}{9}\). This is the simplest form without any parentheses or further reduction possible.

Key concepts

Fractions

Fractions are a way to represent numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the top number and represents how many parts we have, while the denominator is the bottom number and tells us how many parts make up a whole.
To understand fractions better, it's important to know they express division. For example, the fraction \( \frac{1}{3} \) means one part out of three equal parts.

Working with fractions often involves operations like addition, subtraction, multiplication, and division. Here are some basic strategies:

• Addition/Subtraction: To combine fractions, they need a common denominator. For instance, when simplifying \( \frac{1}{3} \) and \( \frac{1}{5} \), we find a common denominator of 15 to rewrite them as \( \frac{5}{15} \) and \( \frac{3}{15} \) respectively.


• Multiplication: Simply multiply the numerators together and the denominators together. For example, multiplying \(-\frac{1}{3} \times \frac{1}{3} = \frac{-1}{9}\).


• Division: To divide by a fraction, multiply by its reciprocal. If you need to divide by \(\frac{2}{5}\), you multiply by \(\frac{5}{2}\).

Mastering these operations is crucial to dealing with fractions in algebra.

Simplification

Simplification in algebra helps to make expressions more manageable and easier to understand. It involves reducing expressions to their simplest form without changing their value.

In the given exercise, simplification is critical. Let's break down the main steps you typically follow:

• Start by handling any operations inside parentheses. In this case, the smallest set of parentheses \( \left(\frac{1}{3} - \frac{1}{5}\right) \) is simplified first.


• Next, continue simplifying step by step by performing arithmetic operations. This means applying rules for addition, subtraction, multiplication, or division while keeping the expressions within combined or simplified form.


• Reduce fractions whenever possible. For instance, during the solution process, the fraction \( \frac{2}{30} \) simplifies to \( \frac{1}{15} \).

By applying these steps systematically, you can simplify even complex algebraic expressions down to a basic, understandable form.

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators like addition or subtraction that represent a particular value.

The use of mathematical expressions allows for compact and efficient communication of mathematical ideas. In algebra, these expressions can be transformed using various rules and properties.

To solve or simplify mathematical expressions, it's essential to:

• Understand the structure: Look at how numbers and operations are arranged. Operations inside parentheses are performed first, following the order of operations (PEMDAS/BODMAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).


• Manage operations: Pay close attention to signs (positive or negative) associated with numbers, especially during operations like distribution. For instance, distributing \(-\frac{1}{2}\) correctly across a fractional expression is vital.


• Simplify step-by-step: Handle one arithmetic operation at a time, checking each step thoroughly to ensure accuracy. This is particularly important when working with nested operations or multiple layers of fractions.

By learning how to effectively manage and simplify mathematical expressions, problem-solving becomes more straightforward, and concepts become much clearer.