Problem 22 - Beginning and Intermediate Algebra

Question

Simplify. $$3 y^{4} \cdot 4 y^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(12y^{7}\).

Step 1: Apply the distributive property

We need to multiply the constants and keep the variables with their exponents: \((3 \cdot 4)(y^{4} \cdot y^{3})\).

Step 2: Multiply the constants

Now we multiply the constants 3 and 4: \(3 \cdot 4 = 12\).

Step 3: Apply the properties of exponents

When multiplying two exponential expressions with the same base, we add their exponents: \(y^{4} \cdot y^{3} = y^{4+3}\).

Step 4: Simplifying the expression

Now we will combine the results of steps 2 and 3: \(12y^{4+3}\).

Step 5: Simplify the exponent

Add the exponents: \(4 + 3 = 7\).

Step 6: Write the final answer

Our final answer is \(12y^{7}\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to simplify expressions by distributing multiplication over addition or subtraction. In the context of simplifying expressions like \(3y^4 \cdot 4y^3\), this property ensures that we can multiply the constants and the variables separately. This step is crucial in helping us organize our calculations. By applying the distributive property, we can rewrite the expression as \((3 \cdot 4)(y^4 \cdot y^3)\). Here, we group the constants \(3\) and \(4\) together, and the variable parts \(y^4\) and \(y^3\) together. This preparatory action paves the way for the following steps of simplification.

Properties of Exponents

Understanding the properties of exponents is key to simplifying algebraic expressions involving powers. When you multiply terms with the same base, you can add the exponents, according to the rule: \(a^m \cdot a^n = a^{m+n}\). In our example \(3y^4 \cdot 4y^3\), we focus on the variables part \(y^4 \cdot y^3\). Here, 'y' is the base, and we have two exponents \(4\) and \(3\) associated with it. By adding these exponents, \(4 + 3\), we simplify the expression to \(y^7\). This rule simplifies expressions by reducing multiple exponent terms into a single term, making calculations more manageable.

Multiplying Constants

Multiplying constants is a straightforward yet critical step in algebraic simplification. In mathematical terms, constants are numbers without any variables. When multiplying such constants, like in our example where we have \(3\) and \(4\), you simply multiply them as you would normal numbers: \(3 \cdot 4 = 12\). The product of these constants will be multiplied with the variable part of your expression, contributing to your final simplified expression. Consistently paying attention to this step ensures the numerical accuracy of your final result.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the basic building blocks in algebra. Simplifying these expressions is vital, as it offers a cleaner and more understandable representation of the problem. Take an expression like \(3y^4 \cdot 4y^3\). By breaking it down using algebra principles, like the distributive property and properties of exponents, we can simplify it to \(12y^7\). Understanding how components work together in an algebraic expression is essential: variables (like \(y\)), exponents (\(4\) and \(3\)), and constants (\(3\) and \(4\)). Simplification helps in performing calculations and solving equations more efficiently.