Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\left(6 y^{2}\right)\left(2 y^{3}\right)^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(48 y^{11}\).

Step 1: Distribute Exponents

In this step, distribute the exponent 3 to \(2 y^{3}\). It applies to both base and exponent. So, \( (2 y^{3})^{3} = 2^{3} \cdot (y^{3})^{3} \).

Step 2: Apply the Power Rule

According to the power rule, \((y^{3})^{3} = y^{3 \cdot 3}\). Thus, \( (2 y^{3})^{3} = 2^{3} \cdot y^{9}\). Calculate \(2^{3}\) to get \(8\). Thus, \( (2 y^{3})^{3} = 8 y^{9}\).

Step 3: Carry Out the Multiplication

We now multiply the simplified component from step 2 by \(6 y^{2}\). That is \( 6 y^{2} \cdot 8 y^{9} = 6 \cdot 8 \cdot y^{2+9}\) since when multiplying variables, we add the exponents.

Step 4: Simplifying multiplication

Carrying out the multiplication, we get \(48 y^{11}\) as the simplified expression.

Key Concepts

Distribute Exponents

Understanding how to correctly distribute exponents across terms in an algebraic expression is vital to simplifying complex expressions. When you see an expression like \( (ab)^n \), it means you need to raise both the base \(a\) and \(b\) to the power of \(n\).

Applying this concept, when given \( (2y^3)^3 \), you must distribute the exponent of 3 to both 2 and \(y^3\) separately. This can be also viewed as applying the exponent to every factor inside the parentheses. This process simplifies \( (2y^3)^3 \), to \( 2^3(y^3)^3 \). It breaks down the problem into smaller, more manageable parts that can be solved sequentially.

Power Rule

The Power Rule is a fundamental rule in algebra that deals with raising a power to another power. If you have an expression like \( (x^m)^n \), the Power Rule simplifies it to \( x^{m \cdot n} \).

Let's see this rule in action with our given expression \( (y^3)^3 \). According to the Power Rule, rather than multiplying \(y^3 \), three times, you simply multiply the exponents: \( 3 \cdot 3 \). Our expression, therefore, simplifies to \( y^9 \). This step significantly reduces the complexity of solving by providing a quick way to handle exponents raised to exponents.

Multiplying Polynomial Expressions

Multiplying polynomial expressions is just like multiplying simpler, numerical expressions, but it involves variables as well. When you multiply terms like \(6y^2\) and \(8y^9\), you multiply the coefficients (6 and 8), and add the exponents of like bases (2 and 9 for \(y\)).

Thus, the multiplication \(6y^2 \cdot 8y^9\) becomes \(48y^{2+9}\), which simplifies to \(48y^{11}\). Remember, when multiplying terms with the same base, you add their exponents. It's like saying, 'How many \(y\)'s do I have in total if I multiply these two terms together?' This systematic approach helps track each part of the polynomial during multiplication, ensuring the expression is accurately simplified.