Problem 44 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(-3 x^{4} y^{6}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified expression of \((-3 x^{4} y^{6})^{3}\) is -27\(x^{12}y^{18}\).

Step 1: Identify the Base Terms and Powers

Here, we have three base terms: -3, \(x^{4}\), and \(y^{6}\). These are all being raised to the power of 3.

Step 2: Apply the Power Rule

According to the power rule, when a power is raised to another power, powers are multiplied. So, we apply the rule to each base term separately: \((-3)^{3}\), \((x^{4})^{3}\), and \((y^{6})^{3}\).

Step 3: Simplify Each Term

The term \((-3)^{3}\) equals -27, \((x^{4})^{3}\) equals \(x^{12}\) since we multiply 4 by 3, and similarly \((y^{6})^{3}\) becomes \(y^{18}\).

Key Concepts

Understanding the Power Rule

The power rule is a guideline in exponential expressions. It helps us simplify expressions where an exponent is raised to another exponent.

When you see something like \((a^m)^n\), this tells us to multiply the exponents: \((m \times n)\). Applying this concept simplifies the expression to \(a^{m \times n}\).

If you look at the exercise, \((-3 x^{4} y^{6})^{3}\), apply the rule to each part separately:

• The constant part, \((-3)^3\), which equals -27.
• The variable \(x\) with \(x^4\) raised to the 3rd power becomes \(x^{4 \times 3} = x^{12}\).
• The variable \(y\) follows similarly: \(y^{6 \times 3} = y^{18}\).

This rule ensures your calculations remain accurate and straightforward.

Simplifying Exponential Expressions

Simplifying expressions is about making them easier to understand and work with, without changing their value.

After applying the power rule, combine all simplified parts to get the final form.

• For \((-3)^3\), you calculate the power of the negative integer, resulting in -27.
• For the variables \(x\) and \(y\), using the power rule means multiplying the exponents.

This gives us direct values for each term. The simplified expression, \(-27x^{12}y^{18}\), is neater and maintains its original value.

Algebraic Expressions Made Simple

Algebraic expressions, like \((-3 x^{4} y^{6})^{3}\), contain constants, variables, and exponents.

Understanding these components helps simplify complex problems.

• Constants: Numbers with a fixed value, such as -3 in our expression.
• Variables: Symbols like \(x\) or \(y\) that can represent different values.
• Exponents: These indicate how many times a number or variable is multiplied by itself.

By identifying each, and knowing how they interact (like using the power rule), you can easily transform and simplify expressions.

This foundational understanding empowers you to handle more complex algebra with confidence.