Problem 12 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(x y^{5} \cdot x y^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^2 y^7\).

Step 1: Write the Original Expression

The original expression given is \( x y^{5} \cdot x y^{2} \). Our task is to simplify this expression by applying the properties of exponents.

Step 2: Apply the Commutative and Associative Properties

Reorder the terms by using the commutative property of multiplication, which allows us to rearrange factors. Group the like terms together: \( (x \cdot x) \cdot (y^{5} \cdot y^{2}) \).

Step 3: Combine Like Bases Using the Product of Powers Property

Apply the property of exponents that states \(a^m \cdot a^n = a^{m+n}\). For \(x\), this becomes \(x \cdot x = x^1 \cdot x^1 = x^{1+1} = x^2\). For \(y\), it becomes \(y^{5} \cdot y^{2} = y^{5+2} = y^{7}\).

Step 4: Write the Simplified Expression

Combine the results from the previous step to establish the simplified expression: \(x^2 y^7\).

Key Concepts

Properties of Exponents

Exponents are mathematical expressions involving a base and an exponent. Understanding their properties can make complex calculations much simpler. One essential property is the "Product of Powers Property," which helps us handle terms with the same base.

• **Product of Powers**: If you have the same base multiplied together with different exponents, you add the exponents. For instance, with bases like in our exercise, if you have \(a^m \cdot a^n\), this simplifies to \(a^{m+n}\).
• **Zero Exponent**: Any base to the zero power, except zero itself, is 1, i.e., \(a^0 = 1\).
• **Negative Exponent**: A negative exponent represents the reciprocal of the base raised to the positive version of that exponent, e.g., \(a^{-n} = \frac{1}{a^n}\).

Understanding these properties gives you the tools to simplify expressions efficiently.

Commutative Property

The commutative property is a fundamental principle in algebra. It lets you rearrange the terms, which is useful when working with expressions.

• **Addition**: \(a + b = b + a\)
• **Multiplication**: \(a \cdot b = b \cdot a\)

By enabling us to rearrange terms, this property allows for easier simplification of algebraic expressions. In our exercise, it allowed us to group like bases together (e.g., arranging \(xy^5 \cdot xy^2\) as \((x \cdot x) \cdot (y^5 \cdot y^2)\)). This step is crucial before applying other properties like those of exponents.

Associative Property

The associative property goes hand-in-hand with the commutative property. It allows us to group different combinations of numbers in a problem without affecting the result.

• **Addition**: \((a + b) + c = a + (b + c)\)
• **Multiplication**: \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)

This property is particularly useful when simplifying expressions by grouping terms. In our example, once the terms were rearranged, the associative property allowed us to regroup to \((x \cdot x) \cdot (y^5 \cdot y^2)\), allowing direct application of the "Product of Powers Property." This grouping ensures efficiency and correctness in simplification.

Product of Powers Property

The product of powers is a handy exponent rule that simplifies expressions with like bases. It states that multiplying two powers with the same base allows you to add the exponents.
For example, the expression \(a^m \cdot a^n\) becomes \(a^{m+n}\). This property streamlines expressions by consolidating terms, reducing what appears to be a tangled mess into simpler, more manageable terms.
In our exercise, after employing the commutative and associative properties to rearrange the expression to \((x \cdot x) \cdot (y^5 \cdot y^2)\), we applied the product of powers property:

• \(x \cdot x = x^{1+1} = x^2\)
• \(y^5 \cdot y^2 = y^{5+2} = y^7\)

Thus, our simplified expression becomes \(x^2 y^7\), demonstrating the efficacy of exponent rule applications.