Problem 38 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(4 x^{2} y^{3}\right)^{3}$$

Step-by-Step Solution

Answer

The expression simplified is \(64x^6y^9\).

Step 1: Identify each part of the monomial

The monomial given is \(4x^2y^3\). It consists of a coefficient, 4, and variables \(x\) and \(y\) with their respective exponents, 2 and 3.

Step 2: Apply the power to the coefficient

When the entire monomial \((4x^2y^3)^3\) is raised to the power of 3, the coefficient 4 is raised to the power of 3. Calculate \(4^3 = 4 \times 4 \times 4 = 64\).

Step 3: Apply the power to the variable \(x\)

Raise \(x^2\) to the power of 3. Use the rule \((a^m)^n = a^{m \cdot n}\). Thus, \((x^2)^3 = x^{2 \times 3} = x^6\).

Step 4: Apply the power to the variable \(y\)

Raise \(y^3\) to the power of 3. Similarly using \((a^m)^n = a^{m \cdot n}\), calculate \((y^3)^3 = y^{3 \times 3} = y^9\).

Step 5: Combine the results

Combine the results from the previous steps to get the final expression: \(64x^6y^9\).

Key Concepts

Monomials

A monomial is a type of algebraic expression that consists of a single term. In mathematics, this term might include numbers, variables, or both. For example, the expression \(4x^2y^3\) is a monomial. It consists of:

• A coefficient: In this case, 4, which is the numeric part of the monomial.
• Variables: Here, the variables are \(x\) and \(y\).
• Exponents: Each variable is raised to an exponent. In our example, \(x\) is raised to the power of 2, and \(y\) is raised to the power of 3.

This structure helps in performing operations such as multiplication, division, and raising a monomial to a power with ease. Understanding each part of a monomial allows us to manipulate it correctly according to algebraic rules.

Power Rule

The power rule is a basic principle in algebra that helps simplify expressions like monomials when raised to a power. The power rule states that when raising a power to another power, you multiply the exponents. This can be written as \((a^m)^n = a^{m \cdot n}\).Applying this to our example \((4x^2y^3)^3\), we get several calculations:

• The coefficient 4 is raised to the third power: \(4^3 = 4 \times 4 \times 4 = 64\).
• The variable \(x^2\) is raised to the third power: \((x^2)^3 = x^{2 \times 3} = x^6\).
• The variable \(y^3\) is raised to the third power: \((y^3)^3 = y^{3 \times 3} = y^9\).

This rule greatly simplifies operations involving powers and makes working with monomials in polynomial expressions very efficient.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. These expressions can be as simple as a single monomial like \(4x^2y^3\) or more complex combinations of multiple terms.Understanding algebraic expressions is crucial in algebra as they form the foundation for equations and functions. Basic components of algebraic expressions include:

• Coefficients, which are the numerical factors in terms.
• Variables, which represent unknown values and can vary.
• Exponents, which indicate the power to which a variable is raised.
• Operators, such as addition, subtraction, multiplication, and division, which relate the terms in the expression.

To manipulate algebraic expressions effectively, it's important to apply rules like the distributive law, combine like terms, and use exponent rules, such as the power rule, to simplify and solve them in equations.