Problem 31 - Algebra and Trigonometry

Question

Simplify each expression. $$ \frac{10(x+y)^{4}}{5(x+y)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2(x+y)\).

Step 1: Identify the Common Factor

Notice that the expression contains the term \((x+y)\) in both the numerator and denominator. Both are raised to powers: 4 in the numerator and 3 in the denominator.

Step 2: Apply the Power Properties

Use the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\). This means \(\frac{(x+y)^4}{(x+y)^3} = (x+y)^{4-3} = (x+y)^{1}\).

Step 3: Simplify the Coefficient

Divide the coefficients: \(\frac{10}{5} = 2\).

Step 4: Combine Results

Put the simplified exponents and coefficients together: \(2(x+y)^{1} = 2(x+y)\).

Key Concepts

Exponent Rules

Exponent rules are essential in algebra for simplifying expressions. They help us manage terms raised to powers efficiently. In our given expression, both numerator and denominator involve the base \((x + y)\). The numerator has the exponent 4 and the denominator 3. When you see terms with similar bases raised to exponents, recall the rule:

• \(\frac{a^m}{a^n} = a^{m-n}\)
• This means you subtract the exponents when dividing.

Applying this rule to the expression \(\frac{(x+y)^4}{(x+y)^3}\) simplifies to \((x+y)^{4-3} = (x+y)^{1}\). Thus, understanding exponent rules allows us to simplify complex terms easily.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. It's about making math look cleaner and easier to understand. In our example:

• The expression starts as \(\frac{10(x+y)^4}{5(x+y)^3}\).
• By using exponent rules and basic arithmetic, the significant part simplified was \(\frac{(x+y)^4}{(x+y)^3}\).
• This reduced to \((x+y)^{1}\).

Simplifying doesn't just involve reducing exponents. It includes collecting like terms and reducing coefficients as well. With practice, these steps become swift and intuitive.

Common Factor

Identifying a common factor is a crucial step in algebra simplification. It involves recognizing like terms or factors that appear in both parts of a fraction. In our exercise:

• Both the numerator and denominator have the factor \((x + y)\).
• This factor appears with different exponents: 4 in the numerator and 3 in the denominator.
• By recognizing this commonality, we could use exponent rules to simplify the expression efficiently.

Spotting common factors quickly can drastically reduce the complexity of algebraic expressions, making them more manageable.

Coefficients in Algebra

Coefficients are numbers that multiply variables or expressions in algebra. They are integral to simplification:

• In the given expression, 10 and 5 are coefficients of \((x + y)^4\) and \((x + y)^3\), respectively.
• Dividing these coefficients \(\frac{10}{5}\) gives us 2, simplifying that part of the fraction.

Simplifying coefficients is a straightforward arithmetic task, but it's crucial to achieving a fully simplified expression. Remember, coefficients carry significant weight in expressions, affecting the magnitude of terms involved.