Problem 44 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \((10 x y)^{2}\)

Step-by-Step Solution

Answer

The simplified expression for \((10 x y)^{2}\) is \(100x^2y^2.\)

Step 1: Recognize the exponent expression rule

First, note that according to the rules of exponents, when a whole product is raised to a power, it can be expressed as each factor to that power, individually. This means that \((10xy)^2\) can be written as \(10^2 * x^2 * y^2\).

Step 2: Calculate \(10^2\)

Next, calculate \(10^2\). The result is 100.

Step 3: Write down the final simplified expression

Finally, combine the results from steps 1 and 2 to get the simplified expression. It results in \(100x^2y^2.\)

Key Concepts

Exponent Expression Rule

Understanding the rules governing exponents is critical for simplifying expressions in algebra. When you come across an expression like \( (10xy)^2 \), the exponent expression rule dictates that you apply the power to each factor of the expression separately. In simpler terms, if you have a product of factors all to a raised power, such as \( (ab)^n \), you can apply the exponent to each factor individually to get \( a^n * b^n \).

For the given expression \( (10xy)^2 \), you would raise each component—10, x, and y—to the second power. This results in \( 10^2 * x^2 * y^2 \), where \( 10^2 \) is a simple numerical calculation. By fully grasping this rule, you can dissect more complex expressions with confidence, applying the same principle to break them down into more manageable parts.

Practical Application of the Rule

When applying the exponent expression rule, it is important to remember that it only applies to multiplication within the parentheses. If you have addition or subtraction, the rule changes, which is a common area where students make mistakes. Stay vigilant in identifying when to apply which rules.

Powers of Monomials

A monomial is an algebraic expression consisting of one term with a non-negative integer exponent, such as \( a^n \), where \( a \) is a constant or a variable. The powers of monomials follow specific patterns when it comes to multiplication and exponentiation.

When you raise a monomial to a power, you're essentially raising each factor in the monomial to that power, as seen in the expression from the exercise \( (10xy)^2 \). It simplifies to \( 100x^2y^2 \), because each factor in the monomial \( 10 \), \( x \) and \( y \) is squared.

Remember Order of Operations

In complex monomials, always keep in mind the order of operations (PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). This means, for instance, that if your monomial has a nested exponent, you deal with the innermost expression first. Understanding the behavior of monomials when they're manipulated is key to mastering algebraic simplification and will be a powerful tool in your mathematics toolkit.

Algebraic Simplification

Algebraic simplification is all about making expressions more manageable and easier to understand or compute. It's a process that requires a firm grasp of arithmetic rules, exponent laws, and properties of equality. When simplifying an expression like \( (10 x y)^{2} \), we performed algebraic simplification by first applying the exponent to each factor and then computing the numerical power.

The simplification process often involves combining like terms, factoring, expanding products, and reducing fractions to their lowest terms. The goal is to transform the expression into its simplest form without changing its value.

Strategies for Simplification

Work systematically, step by step, ensuring no rule is overlooked. Use exponent rules efficiently, factor where possible, and always be on the lookout for common errors, like distributing powers across terms added or subtracted inside parentheses. Simplification can sometimes feel tedious, but it is foundational for solving complex mathematical problems and will serve you well across different areas of mathematics.