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Chapter 10: Problem 113

In the following exercises, simplify each expression. $$\left(10 x^{2} y\right)^{3}$$

Short Answer

Expert verified
1000x^6y^3

Step by step solution

01

Apply the Power Rule

For the expression \(\big(10x^2y\big)^3\), apply the power rule \(\big(a \times b\big)^n = a^n \times b^n\) to distribute the exponent 3 to each term inside the parentheses.
02

Distribute the Exponent

Split the expression into individual components: \(10^3 \times (x^2)^3 \times y^3\).
03

Simplify Each Term

Simplify each component separately. \(10^3 = 1000\), \((x^2)^3 = x^{2 \times 3} = x^6\), and \(y^3 = y^3\).
04

Combine the Simplified Terms

Combine the simplified terms to get the final expression: \(1000x^6y^3\).

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Key Concepts

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

Power Rule in Algebra
The power rule is a fundamental concept in algebra that helps us manage expressions involving exponents. It's defined as \((a \times b)^n = a^n \times b^n\). This means when raising a product to a power, we can distribute the exponent to each factor within the product. Apply the power rule carefully to avoid mistakes in combining the terms. In the given exercise \((10x^2y)^3\), we use the power rule to distribute the exponent 3 to each individual term inside the parentheses: \(10^3 \times (x^2)^3 \times y^3\). This allows us to simplify each component one by one.
Understanding Exponents
Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, \(x^2\) means \(x \times x\), while \(x^3\) means \(x \times x \times x\). When dealing with exponents, specific rules help simplify expressions. Some basic rules include:
  • Product Rule: \(a^m \times a^n = a^{m+n}\)
  • Quotient Rule: \(a^m / a^n = a^{m-n}\)
  • Power Rule: \((a^m)^n = a^{m \times n}\)
In the exercise, \((10x^2y)^3\), the power rule was applied to both numeric and algebraic components: \([(x^2)^3 = x^{2 \times 3} = x^6]\). This illustrates the power rule in handling algebraic expressions.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest forms for easier interpretation and solving. When simplifying, it's essential to follow arithmetic rules and carefully apply operations. Here are key steps:
  • Identify and apply relevant algebraic rules, such as the power rule or product rule.
  • Simplify each part independently before combining them.
  • Combine like terms, when applicable.
In the given exercise, \(10^3 \times (x^2)^3 \times y^3\) was simplified step by step:
  1. Simplify each term individually: \10^3 = 1000, (x^2)^3 = x^6\, and \y^3 = y^3\.
  2. Combine the simplified terms: \1000x^6y^3\.
This method ensures clarity, accuracy, and easier handling of mathematical expressions.

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