Problem 9 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$\left(2 x^{5} y^{4}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(8x^{15}y^{12}\).

Step 1: Apply the Power Rule for Exponents

The Power Rule states that \((a^m)^n = a^{m \cdot n}\). Apply this rule separately to each component of the given expression \((2x^5y^4)^3\).

Step 2: Simplify the Coefficient

First, simplify the coefficient: \((2)^3 = 8\). Therefore, the simplified coefficient is 8.

Step 3: Simplify the Exponent on x

Apply the power rule to the variable \(x\): \((x^5)^3 = x^{5 \times 3} = x^{15}\). Thus, the exponent on \(x\) becomes 15.

Step 4: Simplify the Exponent on y

Apply the power rule to the variable \(y\): \((y^4)^3 = y^{4 \times 3} = y^{12}\). Thus, the exponent on \(y\) becomes 12.

Step 5: Combine the Results

Combine the results from the previous steps to express the simplified version of the original expression: \(8x^{15}y^{12}\).

Key Concepts

Power Rule

The Power Rule is an important tool in algebra that simplifies the process of working with exponents. This rule states that when you raise a power to another power, you can multiply the exponents:

• \((a^m)^n = a^{m \cdot n}\)

In this formula, \(a\) is the base, \(m\) is the first exponent, and \(n\) is the second exponent. The Power Rule helps us to simplify expressions where exponents appear in layers.


For instance, in the expression \((2x^5y^4)^3\), we apply the Power Rule to each component:

• The number 2 is raised to the power of 3: \(2^3\)
• The variable \(x^5\) becomes \((x^5)^3 = x^{5 \times 3} = x^{15}\)
• Similarly, \(y^4\) becomes \((y^4)^3 = y^{4 \times 3} = y^{12}\)

Applying the Power Rule correctly leads to a simplified expression, making calculations easier and more efficient.

Exponents

Exponents are a fundamental concept in algebra, representing repeated multiplication. For example, \(x^5\) means \(x\) is multiplied by itself five times: \(x \times x \times x \times x \times x\). Using exponents lets us express large numbers or complex multiplication more compactly.

The rules of exponents, like the Power Rule, assist us in manipulating these expressions. When dealing with much larger calculations or simplifications, understanding these rules provides a powerful way to work algebraically without expanding each multiplication.

In the exercise, different components use exponents:

• The base 2 is an exponent: \(2^3\) which means \(2 \times 2 \times 2 = 8\).
• Both \(x\) and \(y\) are raised to powers which we simplify using the Power Rule.

Understanding this allows us to handle expressions efficiently. Throughout mathematics, exponents simplify expressions, model growth processes, such as population or radioactive decay, and even describe astronomical distances.

Simplification

Simplification is the process of making an algebraic expression easier to work with by condensing it into its simplest form. This means taking a complex equation and reducing it into one that's more direct while maintaining the equation's inherent relationships.

The simplification process in the given expression \((2x^5y^4)^3\) involves several fundamental steps:

• First, use the Power Rule to consolidate exponents and make the equation less complicated.
• Next, calculate the numerical coefficients by raising them to the given power: for 2, we calculate \(2^3=8\).
• Execute the Power Rule on each variable, \(x\) and \(y\), to simplify their powers to \(x^{15}\) and \(y^{12}\) respectively.
• Lastly, weave together these components to express the simplified version: \(8x^{15}y^{12}\).

Each reduction step aids understanding and assists in applying these principles to more complex problems. Practicing simplification hones your ability to handle more advanced algebraic expressions seamlessly.