Problem 47 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(9 x y^{4}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(81x^2y^8\).

Step 1: Understand the Problem

We need to raise the monomial \((9xy^4)^2\) to the power indicated, which is 2.

Step 2: Apply the Power Rule

The power rule for exponents states that \((a^m)^n = a^{m \times n}\). This means we need to apply the exponent 2 to both the coefficient and the variables inside the parentheses.

Step 3: Raise the Coefficient to the Power

The coefficient inside the parentheses is 9. We apply the exponent 2: \(9^2 = 81\). This gives us the coefficient for our result.

Step 4: Raise Each Variable to the Power

The variables inside the parentheses are \(x\) and \(y^4\). Raise each to the power of 2: \(x^2\) and \((y^4)^2\). Using the power rule, \((y^m)^n = y^{m \times n}\), so \((y^4)^2 = y^{8}\).

Step 5: Write the Final Expression

Combine the expanded coefficient and variables together: \(81x^2y^8\).

Key Concepts

Power Rule for Exponents

The power rule for exponents is an essential concept in algebra that helps simplify expressions. It tells us that when you have a power raised to another power, you multiply the exponents. This can be written as

• For any number or variable \((a^m)^n = a^{m \times n}\)

For example, if you have \((y^4)^2\),you multiply the exponents: \(4 \times 2 = 8\).So, \((y^4)^2 = y^8\).
This rule is particularly handy when dealing with polynomials, as it can transform complicated expressions into simpler ones.Simply focus on applying the power to each part of the expression inside the parentheses separately. Whether you have a number, a variable, or both, use the same method.When raising a product to an exponent, keep in mind:

• Apply the exponent to each factor inside the product.

This approach allows you to handle each component step-by-step for clarity and precision.

Raising Monomials

Raising monomials involves applying exponents to expressions consisting of a single term. In our exercise, the monomial is \(9xy^4\).When raising this to a power, separate the expression into its components: the coefficient and the variables. For monomials:

• Identify each factor: numbers (coefficients) and variables with their exponents.
• Raise each part to the given power as per the exercise requirements.

For example, raise the coefficient 9 to the power of 2: \(9^2 = 81\).
Next, apply the power rule to each variable:

• For \(x\): \(x^2\).
• For \(y^4\): \((y^4)^2 = y^{8}\).

Putting it all together, you get \(81x^2y^8\).Each step involves precise application of the power on each component, providing a clear final result.

Polynomial Expressions

Polynomial expressions consist of terms which can be monomials themselves or combinations of several monomials. In this context, a monomial is a component of a polynomial. Polynomials can look complex, but breaking them down into simpler parts makes them manageable.
When working with polynomials involving powers, focus on each individual term. For example, with a simple expression like \((9xy^4)^2\),each monomial term needs to be attended to, as seen in previous sections.

• Monomials are made of numbers (coefficients) and variables with exponents attached.
• Use rules like the power rule to systematically simplify each term.

Understanding how individual terms interact under operations is key to manipulating polynomials effectively. Polynomials might involve additions or subtractions between terms, but the strategy remains similar: tackle each monomial separately, apply rules like the power rule, and then rewrite the full polynomial expression.
Breaking down complex expressions into manageable parts is the best approach to handling polynomials with powers. This builds both skills and confidence in working with larger expressions.