Problem 27 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(x^{9} y\right)\left(x^{10} y^{5}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^{19} y^{6} \).

Step 1: Identify the Product Rule

The product rule of exponents states that when you multiply two expressions with the same base, you add their exponents. The rule can be written as \( a^m \times a^n = a^{m+n} \). In this problem, you'll need to apply this rule separately to the bases \( x \) and \( y \).

Step 2: Apply the Product Rule to the Base x

Identify the base \( x \) terms in the expression: \( x^9 \) and \( x^{10} \). Using the product rule, you add their exponents: \( 9 + 10 = 19 \). Thus, the simplified form for \( x \) is \( x^{19} \).

Step 3: Apply the Product Rule to the Base y

Identify the base \( y \) terms in the expression: \( y^{1} \) (because \( y = y^1 \)) and \( y^{5} \). Using the product rule, add their exponents: \( 1 + 5 = 6 \). Thus, the simplified form for \( y \) is \( y^{6} \).

Step 4: Combine the Results

Combine the simplified terms for bases \( x \) and \( y \): \( x^{19} \cdot y^{6} \). Therefore, the expression \( \left(x^{9} y\right)\left(x^{10} y^{5}\right) \) simplifies to \( x^{19} y^{6} \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. They are used to represent mathematical relationships and problems in a concise form. In simple terms, an algebraic expression might look like a combination of constants (like 3 or 9), variables (like \( x \) or \( y \)), and mathematical operations (such as addition or multiplication).

For example, the expression \( x^9 y \) is an algebraic expression where \( x^9 \) and \( y^1 \) are terms, and the entire expression indicates multiplication between these terms. Understanding how to manipulate these expressions is crucial as they form the foundation of algebra and pre-calculus.

When working with algebraic expressions, you'll often need to expand, simplify, or evaluate them, which involves using various mathematical principles and rules, such as the product rule of exponents, to get a simpler or more useful form of the original expression.

Simplifying Expressions

Simplifying expressions is a key concept in algebra, allowing you to reduce complex problems into simpler, more manageable ones. It involves manipulating an algebraic expression following certain rules to combine like terms and eliminate unnecessary components.

When you simplify an expression, you perform operations like:

• Combining like terms by adding or subtracting coefficients
• Using exponent laws to manage powers
• Canceling common factors in fractions

Simplifying expressions helps in solving equations, conducting algebraic manipulations, and modeling real-world situations more efficiently.

In the exercise \( (x^9 y)(x^{10} y^5) \), the simplification involves using the product rule of exponents. By applying the rule, you combine the bases while adjusting their exponents. This results in a more straightforward expression: \( x^{19} y^6 \). Simplification helps make expressions easier to read and more useful for further calculations.

Exponents Laws

Exponents laws are essential rules used in algebra to manipulate expressions involving powers. These laws help when you're dealing with repeated multiplication of the same number or variable, making calculations less cumbersome.

The key rules are:

• Product Rule: \( a^m \times a^n = a^{m+n} \) - Used when multiplying two powers with the same base.
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \) - Applied when dividing powers of the same base.
• Power Rule: \( (a^m)^n = a^{m \cdot n} \) - Used when raising a power to an exponent.

These rules simplify calculations and help rearrange expressions into simpler forms. The product rule, as used in the exercise \( (x^9 y)(x^{10} y^5) \), requires adding exponents of like bases to consolidate terms, resulting in \( x^{19} y^6 \). Understanding these laws is vital for mastering algebra and for solving more complex math problems efficiently.