Problem 41 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-x^{4} y^{5}\right)^{4}$$

Step-by-Step Solution

Answer

Apply the power rule to both terms.

Step 1: Apply the Power of a Power Rule

The expression is \(-x^4 y^5\) raised to the 4th power. Start with the "power of a power" rule, which states that \( (a^m)^n = a^{m imes n} \).

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

Apply the power rule to both terms.

Key Concepts

Power of a Power Rule

The "power of a power rule" is a fundamental concept in algebra and is a critical tool when dealing with exponents in expressions. This rule simplifies raising an exponent to another power by multiplying the exponents together. Here is how the rule looks:

• Given an expression \( (a^m)^n \), the power of a power rule states that you can multiply the exponents: \( a^{m \times n} \).

Let's consider the exercise monomial: - The expression is \( (-x^4 y^5)^4 \). - Applying the power of a power rule involves multiplying each exponent inside the brackets by 4, the exponent of the entire monomial. - So, the exponents apply to both parts: \( x^4 \) becomes \( x^{4 \times 4} = x^{16} \) and \( y^5 \) becomes \( y^{5 \times 4} = y^{20} \). For this case, remember to apply the power to both variables inside the parenthesis. This clarity makes applying this rule straightforward and handy, regardless of which monomial you're dealing with.

Exponents

Exponents are the backbone of algebra, representing repeated multiplication. The notation \( a^n \) where \( a \) is the base and \( n \) is the exponent, indicates that \( a \) is multiplied by itself \( n \) times.Here’s a breakdown of the key concepts:

• The base (e.g., \( x \) or \( y \) in our monomial) is the number or variable being multiplied.
• The exponent shows how many times the base is used as a factor.
• For example, in \( x^4 \): - The base is \( x \). - The exponent is \( 4 \), meaning \( x \times x \times x \times x \).

In the given monomial \( (-x^4 y^5)^4 \), each component has its own exponent which dictates the rule when simplified. Understanding exponents helps us simplify expressions and perform operations like multiplication and division faster.Practice makes perfect for mastering the use of exponents. Be sure to apply them methodically whenever they appear in an expression.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients. They involve operations of addition, subtraction, and multiplication, which can include powers of the variables.Let's understand polynomials with our example:- A polynomial can be as simple as a single monomial: \(-x^4 y^5\). - Each term is a monomial, which is a product of a constant and variables raised to powers. In dealing with polynomials:

• Recognize each term and its degree, which is the sum of exponents of the variables in the term.
• Simplifying a polynomial often involves using rules like the power of a power rule.

When a polynomial is raised to a higher power, each term within it must be calculated accurately to maintain balance in the expression.Understanding polynomials is key to advancing in algebra, as they form elements for modeling real-world problems and solving equations efficiently.