Rewrite The Expression As A Simplified Expression Containing One Term

Rewriting Expressions: Simplifying to a Single Term

Simplifying mathematical expressions is a fundamental skill in algebra and beyond. The ability to condense complex expressions into a single, equivalent term is crucial for problem-solving, understanding relationships between variables, and making calculations more efficient. This article explores various techniques for rewriting expressions, focusing on the goal of achieving a single-term simplification. We'll cover a range of strategies, from combining like terms to applying the distributive property and factoring.

Understanding the Basics: Like Terms and the Distributive Property

Before delving into complex examples, let's solidify our understanding of two essential concepts: like terms and the distributive property. These are the cornerstones of simplifying expressions.

Like Terms: The Foundation of Simplification

Like terms are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms because they both have the variable x raised to the power of 1. However, 3x and 3x² are not like terms because the powers of x differ. Similarly, 2xy and 7xy are like terms, but 2xy and 2x are not.

Combining like terms involves adding or subtracting their coefficients (the numbers in front of the variables). For example:

3x + 5x = 8x

2xy - 7xy = -5xy

4a²b + 9a²b = 13a²b

This simple process forms the basis for many simplification strategies.

The Distributive Property: Unlocking Simplification

The distributive property states that a(b + c) = ab + ac. This means we can distribute a term multiplied by a sum or difference across the terms within the parentheses. The reverse process, factoring, is also crucial for simplifying expressions.

For instance:

2(x + 3) = 2x + 6

5(2y - 4) = 10y - 20

This property allows us to expand or collapse expressions, bringing us closer to a single-term representation.

Techniques for Rewriting Expressions to a Single Term

Now let's explore several techniques used to simplify expressions into a single term. Many problems will require a combination of these methods.

1. Combining Like Terms: The Direct Approach

The most straightforward method is combining like terms. This involves identifying terms with the same variables and exponents and then adding or subtracting their coefficients. Let's look at an example:

Example: Simplify 4x² + 6x - 2x² + 3x + 5

Solution:

1. Identify like terms: 4x² and -2x² are like terms; 6x and 3x are like terms.
2. Combine like terms: (4x² - 2x²) + (6x + 3x) + 5 = 2x² + 9x + 5

While this example doesn't result in a single term, many expressions can be simplified to a single term using only this technique. For instance:

7y - 2y = 5y

This is a simplified expression with a single term.

2. Applying the Distributive Property: Expanding and Factoring

The distributive property is key to simplifying expressions that involve parentheses or factors. We can use it to expand an expression or, conversely, factor it to arrive at a single term.

Example: Simplify 3(2x + 4x²) – 6x²

Solution:

1. Distribute the 3: 6x + 12x² - 6x²
2. Combine like terms: 6x + (12x² - 6x²) = 6x + 6x²

In this case, we didn't get a single term, but the expression is simpler. Sometimes, however, factoring leads directly to a single term:

Example: Simplify 5x(2 + x) - 10x - 5x²

Solution:

1. Distribute 5x: 10x + 5x² - 10x - 5x²
2. Combine like terms: (10x - 10x) + (5x² - 5x²) = 0

This simplifies to a single term: 0.

3. Using Exponent Rules: Simplifying Exponential Expressions

When dealing with exponents, remember these key rules:

• x^a * x^b = x^a+b (When multiplying terms with the same base, add the exponents)
• x^a / x^b = x^a-b (When dividing terms with the same base, subtract the exponents)
• (x^a)^b = x^ab (When raising a power to a power, multiply the exponents)

These rules are crucial for simplifying expressions involving exponents and often lead to single-term solutions.

Example: Simplify (2x²y)³ * (x³y²)

Solution:

1. Apply exponent rules: (8x⁶y³) * (x³y²)
2. Combine like terms (using exponent rules): 8x⁶⁺³y³⁺² = 8x⁹y⁵

This results in a single-term simplified expression.

4. Dealing with Fractions: Finding Common Denominators and Simplifying

Expressions with fractions require finding a common denominator and simplifying the numerator. This can sometimes result in a single-term expression.

Example: Simplify (2/3x) + (4/6x)

Solution:

1. Find a common denominator: The common denominator of 3 and 6 is 6.
2. Rewrite the fractions with the common denominator: (4/6x) + (4/6x)
3. Combine like terms: (8/6x) = (4/3x)

While this doesn't result in a single term in its simplest form, it's a simpler representation.

Example: Simplify (x/2) * (4/x)

Solution:

1. Multiply the numerators and denominators: (4x)/(2x)
2. Simplify by canceling common factors: 2

5. Handling Radicals: Simplifying Square Roots and Higher Roots

Simplifying expressions with radicals requires understanding how to simplify square roots and higher roots. We often use the property √(ab) = √a * √b to simplify complex expressions. Similarly, we can use the property √(a/b) = √a / √b.

Example: Simplify √(16x⁴)

Solution:

1. Simplify using the properties of radicals: √16 * √x⁴ = 4x²

This yields a single term.

Advanced Techniques and Considerations

While the methods above cover many scenarios, some complex expressions may require more sophisticated techniques, such as completing the square or using the quadratic formula, before a single-term simplification is possible. These advanced methods are often used to solve quadratic equations and are beyond the scope of this basic introduction.

Conclusion

Rewriting expressions to a single term is a valuable skill in algebra and mathematics generally. Mastering the techniques of combining like terms, utilizing the distributive property, applying exponent rules, handling fractions and radicals, and strategically choosing the correct path, will significantly improve your mathematical problem-solving abilities. Remember that practice is key; the more you work with these techniques, the more confident and efficient you'll become in simplifying complex expressions to their most concise forms. Remember to always double-check your work and consider the context of the problem to ensure your simplified expression is valid and meaningful within the given constraints.