Problem 45 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (x y)^{4}\left(x^{2} y^{4}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \((x y)^{4}(x^{2} y^{4})\) Answer: \(x^{6} y^{8}\)

Step 1: Identify the expression we need to simplify

The expression we need to simplify is: $$ (x y)^{4}\left(x^{2} y^{4}\right) $$

Step 2: Apply the power rule for exponent \((a b)^n = a^n b^n\)

When we raise a product to a power, we can apply the power to each component: \((x y)^{4} = x^4 y^4\). This gives us: $$ x^4 y^4\left(x^{2} y^{4}\right) $$

Step 3: Multiply terms with the same base

When we multiply expressions with the same base, we add their exponents: \(a^m \cdot a^n = a^{m+n}\). So, we will combine \(x\) terms and \(y\) terms separately: $$ x^{4+2} y^{4+4} $$

Step 4: Simplify the exponents

Adding the exponents of each term, we get: $$ x^{6} y^{8} $$ The simplified expression is: \(\boldsymbol{x^{6} y^{8}}\).

Key Concepts

Simplification of Expressions

When we talk about the simplification of expressions in mathematics, especially involving exponents, the goal is to make the expression easier to read and understand. Simplification involves reducing an expression to its simplest form without changing its value or parameters.
For our exercise, we start with an expression \((x y)^{4}(x^{2} y^{4})\). From a simplicity perspective, this means breaking down and compressing it into a more straightforward expression, which in this case is \(x^{6} y^{8}\).
The key to simplifying is knowing and applying specific mathematical principles, such as the power rules and the ability to combine like terms. This ensures that the expression retains its original properties while appearing in its most efficient form.
Simplification is a fundamental skill in algebra that helps in better handling complex expressions, making it easier to perform subsequent operations like solving equations or comparing expressions.

Multiplication of Exponents

The multiplication of exponents comes into play when you encounter expressions involving powers that need to be combined. This can often simplify the process of dealing with an otherwise complicated expression by reducing the number of terms and operations.
In the given exercise, when we see terms like \((x^4 y^4)(x^2 y^4)\), we aim to multiply the like bases. The rule of thumb for multiplying expressions with the same base is to add the exponents together. In mathematical terms, this rule is expressed as \(a^m \cdot a^n = a^{m+n}\).

• Combining \(x\) terms: \(x^4 \cdot x^2 = x^{4+2} = x^6\)
• Combining \(y\) terms: \(y^4 \cdot y^4 = y^{4+4} = y^8\)

This rule makes it straightforward to condense expressions that otherwise might seem complex, transforming them quickly into simpler equivalents and illustrating the power of mathematical rules in making calculations easier.

Natural Numbers in Exponents

Natural numbers are the set of positive integers used widely in mathematics; these include numbers such as 1, 2, 3, and so forth. When we see exponents that are natural numbers, it simplifies the application of power rules because we aren't dealing with fractional or negative terms which can add complexity.
In this context, when exponents are natural numbers, like in our exercise, each step in applying rules becomes straightforward. Raising a term to a natural number exponent implies repeated multiplication. For instance, \(x^4\) means multiplying \(x\) by itself four times: \(x \cdot x \cdot x \cdot x\).
It's important to understand that while natural numbers provide a simpler case scenario, the power rules still hold valid regardless of whether the exponents are whole numbers or not. Each scenario only adapts the application accordingly, ensuring the main rule is respected no matter the type of number involved.