Problem 47 - Algebra 2

Question

Simplify. $$ x^{-3} y^{2}\left(y x^{4}+y^{-1} x^{3}+y^{-2} x^{2}\right) $$

Step-by-Step Solution

Answer

The simplified form is \( x y^3 + y + x^{-1} \).

Step 1: Distribute the terms inside the parentheses

Distribute each term in the parentheses \( y x^4 + y^{-1} x^3 + y^{-2} x^2 \) by multiplying it with \( x^{-3} y^2 \). This gives us:- \( x^{-3} y^2 \times y x^4 = x^{-3} y^3 x^4 = x^{1} y^3 = x y^3 \)- \( x^{-3} y^2 \times y^{-1} x^3 = x^{-3} x^3 y y^{-1} = y \)- \( x^{-3} y^2 \times y^{-2} x^2 = x^{-1} y^0 = x^{-1} \)

Step 2: Combine the results

Combine the results from the distribution in Step 1:\( x y^3 + y + x^{-1} \). This expression is the simplified form of the original problem.

Key Concepts

Exponent Rules

When simplifying expressions with exponents, understanding exponent rules is crucial. These rules help to simplify expressions or solve equations involving powers effectively. Here are a few key rules that are particularly useful:

• Product of Powers Rule: When multiplying two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: When raising an exponent to another exponent, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

In the exercise, these rules are used to combine and simplify the exponents on variables both within and outside the parentheses. For instance, when distributing \( x^{-3} y^2 \) to \( y x^4 \), the exponents are manipulated using the product of powers rule to simplify to \( x^1 y^3 \). Understanding these rules allows you to systematically simplify complex expressions.

Polynomial Distribution

Polynomial distribution is a method used to multiply a single term by each term within a polynomial. It's similar to the distributive property applied to algebraic expressions. Here's a quick breakdown of how this process works:

• Distributive Property: Multiply each term inside the brackets by the term outside. This is done individually for each term inside:
• For example, in \( x^{-3} y^2 (y x^4) \), you multiply \( x^{-3} y^2 \) with every term inside.
• Once every term is distributed and simplified, combine all the results to form a simplified expression.

In the given problem, \( x^{-3} y^2 \) is distributed across every term in \( y x^4 + y^{-1} x^3 + y^{-2} x^2 \). Each multiplication involves applying exponent rules to simplify each term. This distribution helps in breaking down and managing complex expressions more simply.

Negative Exponents

Negative exponents indicate a reciprocal relationship for the base raised to a positive exponent. This concept allows you to shift terms across the fraction line:

• Basic Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \). This means that any number or variable with a negative exponent can be rewritten as a fraction.
• Simplification: To simplify expressions with negative exponents, convert them to fractions and further simplify if possible.

In the initial exercise, exponents such as \( x^{-3} \) and \( y^{-1} \) demonstrate the use of negative exponents. Working through these, turn them into fractions first and apply other exponent rules to further simplify. For instance, \( x^{-1} \) simplifies to \( \frac{1}{x} \), contributing to the final expression of the problem. Recognizing and converting negative exponents is therefore a vital skill in your algebra toolkit.