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Chapter 4: Problem 29

Simplify. Do not leave negative exponents in your answer. $$ \frac{-6 x^{2}}{3 x^{-10}} $$

Short Answer

Expert verified
-2x^{12}

Step by step solution

01

Simplify the Coefficients

Divide the coefficients (-6 and 3). This simplifies to -2 because \(\frac{-6}{3} = -2\).
02

Simplify the Exponents

Apply the properties of exponents. When dividing like bases, subtract the exponents: \(\frac{x^2}{x^{-10}} = x^{2 - (-10)} \).
03

Simplify the Exponent Expression

Subtract the exponents: \(2 - (-10) = 2 + 10 = 12\), so the exponent simplifies to \(x^{12}\).
04

Combine the Simplified Parts

Combine the simplified coefficient with the simplified variable term to get the final answer: \(-2x^{12}\).

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Key Concepts

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

Negative Exponents
Understanding negative exponents is crucial when working with algebraic expressions. A negative exponent indicates that the base should be inverted. That means if you have a term like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). For example, \(x^{-3}\) is the same as \(\frac{1}{x^3}\). Recognizing this simple rule helps in simplifying more complex fractions and algebraic expressions involving exponents.
Simplifying Fractions
Simplifying fractions forms a fundamental part of algebra. It often involves reducing the coefficient to its simplest form and managing the variables according to their exponents. Here, we simplify the fraction \(\frac{-6}{3}\), resulting in \(-2\).
Another key step is handling the exponents. In the problem, \(\frac{x^2}{x^{-10}}\), you subtract the exponents: \(2 - (-10) = 12\). This results in \(x^{12}\).
A simplified fraction often is much easier to work with and understand.
Properties of Exponents
The properties of exponents help us understand how to manipulate and simplify expressions. Key properties include:
  • Product of powers: \(a^m \cdot a^n = a^{m+n}\).
  • Power of a power: \((a^m)^n = a^{mn}\).
  • Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Negative exponent: \(a^{-n} = \frac{1}{a^n}\).
In the exercise, we use the quotient of powers property: \frac{x^2}{x^{-10}} = x^{2-(-10)}\. This results in \(x^{12}\). Using these properties makes it easier to simplify and combine terms.
Algebraic Expressions
Algebraic expressions consist of numbers and variables combined by arithmetic operations. Simplifying these involves three main steps:
  • Simplify coefficients
  • Apply exponent rules
  • Combine simplified parts
In the given exercise, first simplify the coefficients \(\frac{-6}{3} = -2\). Next, handle the exponents \((2 - (-10) = 12)\), resulting in \(x^{12}\). Finally, put it all together to get the simplified expression: \(-2x^{12}\).
Mastering these steps will make dealing with more complicated algebraic expressions much smoother.

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