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Chapter 1: Problem 74

Simplify. $$ \frac{\left(4 x^{3} y\right)^{2}}{8 x^{2} y^{3}} $$

Short Answer

Expert verified
\(\frac{2x^4}{y}\)

Step by step solution

01

Expand the exponent in the numerator

The given expression is \( \frac{(4x^3y)^2}{8x^2y^3} \). First, expand the squared term in the numerator: \((4x^3y)^2 = 4^2(x^3)^2(y)^2\). This simplifies to \(16x^6y^2\).
02

Simplify the Fraction

The expression is now \( \frac{16x^6y^2}{8x^2y^3} \). Simplify this by dividing both the numerical coefficients and the variables separately. Divide the coefficients: \( \frac{16}{8} = 2 \).
03

Simplify the variables in the fraction

For the variable \(x\), apply the quotient of powers rule: \(x^{6-2} = x^4\). For the variable \(y\), apply the same rule: \(y^{2-3} = y^{-1}\), which can be rewritten as \( \frac{1}{y} \).
04

Combine the results

Combining the simplified parts from previous steps, the simplified expression is \(2x^4 \cdot \frac{1}{y} = \frac{2x^4}{y}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions means making them easier to work with or understand by reducing them to their simplest form. Algebraic fractions like \( \frac{(4x^3y)^2}{8x^2y^3} \) can often be complex at first glance. To simplify such expressions, we start with expanding powers and then divide the terms.
  • Identify parts of the expression that can be simplified: Look for powers, coefficients, and variables both in the numerator and the denominator.
  • Break down complex powers: As shown, \((4x^3y)^2\) was first expanded to \(16x^6y^2\).
  • Divide numerical coefficients separately from variables: We've taken \(\frac{16}{8}\), which simplified the fraction to \(2\).
With practice, simplifying expressions becomes an intuitive process, allowing for a clear and concise representation of the algebraic terms.
Quotient of Powers Rule
The quotient of powers rule is a key principle in algebra that helps when dividing variables with the same base. It states that when dividing like bases with exponents, you simply subtract the exponents. This rule is expressed as:
  • \( \frac{a^m}{a^n} = a^{m-n} \)
Let's see it in action with our expression:
  • For the term \(x\), we have \(x^6\) in the numerator and \(x^2\) in the denominator. Applying the rule gives us \(x^{6-2} = x^4\).
  • For \(y\), do the same: \(y^{2-3} = y^{-1}\), which denotes the reciprocal, \( \frac{1}{y} \).
This rule is incredibly useful because it reduces complex expressions into simpler forms, making calculations easier and faster.
Exponentiation
Exponentiation is a mathematical operation involving numbers or variables raised to a power. When you encounter a term like \((4x^3y)^2\), exponentiation dictates how to expand this term:
  • Each component inside the parentheses is raised to the power outside, so \((4x^3y)^2\) becomes \(4^2 \times (x^3)^2 \times y^2\).
  • This results in \(16x^6y^2\), as each part is individually raised to the power.
Understanding exponentiation helps greatly when dealing with algebraic expressions because it provides a systematic way to handle powers. Couple this with other rules like simplification and the quotient of powers, and you can efficiently manage even the most complex algebraic fractions.

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Most popular questions from this chapter

True or False: A vertical line has slope \(0 .\)

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