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Chapter 10: Problem 116

In the following exercises, simplify each expression. $$\left(-10 u^{2} v^{4}\right)^{3}$$

Short Answer

Expert verified
-1000 u^{6} v^{12}

Step by step solution

01

- Identify the expression

The given expression is \(-10 u^{2} v^{4}\) raised to the power of 3: \(\big(-10 u^{2} v^{4}\big)^{3}\).
02

- Apply the power rule

Use the power rule \(a^m\)^n = a^{m \times n}\ to simplify each term inside the parentheses: \((-10)^{3} (u^{2})^{3} (v^{4})^{3}\).
03

- Simplify each term

Simplify each part individually: \((-10)^{3} = -1000\), \(u^{2 \times 3} = u^{6}\), and \(v^{4 \times 3} = v^{12}\).
04

- Combine the simplified terms

Combine the results from the previous step to get the simplified expression: \(-1000 u^{6} v^{12}\).

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Key Concepts

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

power rule in exponents
Understanding the power rule in exponents is crucial for simplifying expressions involving powers. The power rule states that when you have a base raised to an exponent, and then that entire expression is raised to another exponent, you can multiply the exponents together. This can be formally written as: \((a^m)^n = a^{m \times n}\). This rule helps in breaking down complex exponential expressions into simpler forms.

For example, in the given exercise, \((-10 u^{2} v^{4})^{3}\), we apply the power rule to each term inside the parentheses:
  • The term \((-10)^3\) becomes \(-1000\).
  • The term \(u^2)^{3}\) becomes \(u^{6}\).
  • The term \(v^4)^{3}\) becomes \(v^{12}\).
Applying this rule simplifies our work and helps in systematic problem solving.
simplifying powers
Simplifying powers involves addressing each exponent and base in the expression, making sure to follow algebraic rules. When you see an expression like \((-10 u^{2} v^{4})^{3}\), you need to approach each term separately.

Here's a step-by-step breakdown:
  • Step 1 - Identify the expression: Recognize that each base within the expression \(-10 u^{2} v^{4}\) must be raised to the power of 3.
  • Step 2 - Apply the power rule: Use the power rule \((a^m)^n = a^{m \times n}\).
  • Step 3 - Simplify each term: Calculate each term after raising them to the appropriate power.
  • Step 4 - Combine the terms: Collect the simplified terms for your final answer.

    This step-by-step approach ensures that each part of the expression is addressed correctly, leading to the final simplified form.
algebraic expressions
Algebraic expressions can be simplified by understanding the rules and applying them systematically. Expressions like \-10 u^{2} v^{4})^{3}\ may seem complex, but breaking them down makes it more manageable.

Key points to remember when simplifying such expressions:
  • Always identify the base and its exponent first.
  • Apply the power rule when you have nested exponents.
  • Simplify each term one by one instead of all at once.
  • Finally, combine the simplified terms to form the final expression.

    For instance, in \((-10 u^{2} v^{4})^{3}\), understanding how each term is manipulated by the exponent reduces errors and keeps the process clear.
Keep practicing, and you'll find these steps becoming second nature, making algebra less intimidating and more intuitive.

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