Problem 39 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-2 x^{2} y\right)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is \(-32x^{10}y^{5}\).

Step 1: Apply the Power Rule

The power rule, \[ (a^m)^n = a^{mn}, \] applies to exponents involving multiplication. We'll raise each factor in the monomial to the fifth power: \[ (-2)^5 \cdot (x^2)^5 \cdot y^5. \]

Step 2: Compute Powers Individually

Calculate each component separately:1. \((-2)^5 = -32\) because multiplying five \(-2\)'s results in a negative product.2. \((x^2)^5 = x^{2 \times 5} = x^{10}\) by multiplying the exponents.3. \(y^5\) remains as it is because it's raised to the power of 5.

Step 3: Combine the Results

Multiply the calculated values to find the final result:\[ (-2)^5 \cdot (x^2)^5 \cdot y^5 = -32 \cdot x^{10} \cdot y^5. \]

Step 4: Write the Final Expression

The expression simplified in standard form is:\[ -32x^{10}y^{5}. \]

Key Concepts

Power Rule

Exponentiation is simplified by using the Power Rule, which is a vital concept in algebra. It effectively helps when raising a power to another power. The Power Rule states that \( (a^m)^n = a^{mn} \). This means you multiply the exponents together. Think of it like a simple multiplication of numbers attached to the same base.

• The base remains unchanged.
• The exponents are multiplied.

In the example \( (-2x^2y)^5 \), you apply the Power Rule to each part of the monomial separately. You multiply the exponents of \(x^2\) and \(-2\) by 5. This breaks down to \((-2)^5\), \((x^2)^5\), and \(y^5\). Utilizing the Power Rule is a streamlined way to handle complex expressions, turning complicated operations into manageable steps.

Monomial

When discussing exponentiation, it's essential to understand the building blocks, like monomials. A monomial is an algebraic expression consisting of a single term.

• It may have constants (like numbers).
• It can include variables (like \(x\) or \(y\)), possibly with exponents.

In the exercise \( (-2x^2y) \), this expression is a monomial. It's crucial to see a monomial as a unified block, making operations like exponentiation straightforward. When raising a monomial to a power, each component (constant and variables) is dealt with individually but following the uniform rules of exponentiation. This clarity helps in systematically simplifying and solving complex algebraic problems.

Multiplication of Exponents

Handling expressions with multiple exponents can seem daunting, but the trick is understanding the rule of the Multiplication of Exponents. This rule is integral to simplifying expressions involving multiple layers of exponentiation. Basically, when you have \(a^m \cdot a^n = a^{m+n} \), you add the exponents when you multiply terms with the same base.

• Check if the bases are identical.
• Sum up the exponents beside each base.

For example, \( (x^2)^5 \) becomes \( x^{2 \cdot 5} = x^{10} \). This technique consolidates terms into a simpler form, making assessments or calculations on algebraic expressions more efficient. By multiplying the exponents in steps, you maintain control over the entire expression, ensuring accuracy and precision in your algebraic computations.