Problem 40 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ 5 x^{2}\left(2 y^{3}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Expand the expression: \(5x^2(2y^3)^3\) Answer: \(40x^2y^9\)

Step 1: Apply the power rule

Using the power rule, we distribute the exponent to both terms inside the parentheses: $$ (2y^3)^3 = 2^3y^{3\cdot3} = 8y^9 $$

Step 2: Multiply the terms

Now that we have expanded the exponent, we can multiply the remaining terms together: $$ 5x^2(8y^9) = (5)(8)(x^2)(y^9) $$

Step 3: Simplify the expression

Lastly, we should simplify the expression by combining the constant terms: $$ (5)(8)(x^2)(y^9) = 40x^2y^9 $$ So, the expanded form of the given expression is: $$ 5x^2(2y^3)^3 = 40x^2y^9 $$

Key Concepts

Expansion in Algebraic Expressions

Expansion in algebra often involves removing parentheses by applying mathematical operations. When we expand an algebraic expression like \(5x^2(2y^3)^3\), we are transforming it so all the operations are distributed properly. This usually means eliminating any exponents or nested expressions.

The purpose of expansion is to rewrite the expression without changing its value. This allows for easier manipulation or further operations, such as simplification. Expansion is a critical step in algebra when simplifying complex expressions.

The key takeaway is that expansion involves:

• Removing parentheses
• Distributing any exponents
• Rewriting without altering the expression's overall value

Understanding the Power Rule

The power rule is a fundamental guideline in algebra that simplifies expressions where exponents are present. It states that when you have a power raised to another power, you multiply the exponents. For instance, in the expression \((2y^3)^3\), we apply the power rule to simplify it.

Let's break it down:

• Start with \((2y^3)^3\) and distribute the 3 exponent to both 2 and \(y^3\).
• The expression becomes \(2^3\) and \(y^{3 \cdot 3}\).
• This simplification results in \(8y^9\).

The power rule thus helps to decrease the complexity of expressions, paving the way for further simplification.

It is important to remember that the rule only applies when a power expression is nested within another power, ensuring accuracy in algebraic transformations.

The Art of Simplification

Simplification is the process where we refine an algebraic expression to its most uncomplicated form. It's the last and an essential step in our expansion problem. Once the exponents and multiplication are dealt with, it is time to combine like terms, if feasible, and arrange the expression neatly.

In our example with \((5)(8)(x^2)(y^9)\), the simplification step includes combining the constants \((5)\) and \((8)\) resulting in \(40\).

Simplification involves:

• Combining constants or coefficients
• Ensuring variables and their exponents are clearly represented
• Arranging components to maintain clarity

Ultimately, this step ensures that the expression is concise and easy to interpret. The final expression \(40x^2y^9\) is prime for further analytic or algebraic use, representing an optimally refined version of the original input.