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Chapter 0: Problem 30

Simplify each exponential expression. $$x^{-6} \cdot x^{12}$$

Short Answer

Expert verified
The simplified exponential expression is \(x^{6}\).

Step by step solution

01

Understand the Problem

We have the exponential expression \(x^{-6} \cdot x^{12}\). To simplify this, we need to use the rule of exponents which states that for any base \(x\), the product of \(x^{m} \cdot x^{n}\) equals \(x^{m+n}\).
02

Apply the Rule of Exponents

For our expression \(x^{-6} \cdot x^{12}\), the base \(x\) is the same for both terms. Hence, we can just add the exponent values (-6 and 12) which results in \(x^{-6+12}\).
03

Simplify the Exponent

Solving the equation in the exponent, -6 + 12 equals 6. This gives us the final answer of \(x^{6}\).

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Key Concepts

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

Rule of Exponents
The rule of exponents is a useful mathematical principle when dealing with exponential expressions. Essentially, if you have the same base in multiplication, such as \(x^m \cdot x^n\), you can add their exponents: \(x^{m+n}\). This simplifies expressions where the bases are the same.
  • Same Base: Only applicable when the bases are identical.
  • Add Exponents: Simply sum the exponents to get the new exponent.
  • Example: In our problem \(x^{-6} \cdot x^{12}\), we add \(-6\) and \(12\) to get \(x^6\).
Mastering this rule can significantly reduce the complexity of calculations. You can use it to streamline solutions and avoid lengthy multiplications.
By remembering this simple rule, you ensure that working with exponential expressions becomes efficient and less error-prone.
Simplifying Expressions
Simplifying expressions makes math more manageable by reducing the problem into its simplest form. Here, we applied the rule of exponents to simplify \(x^{-6} \cdot x^{12}\) into \(x^6\).
  • Step-by-Step Simplification: Identify if similar bases exist, then apply the rules to combine them.
  • Reducing Complexity: Once simplified, the expression becomes more straightforward and quicker to work with in further calculations.
  • Expression Aesthetics: Simplification often improves the visual structure, making it easier to spot other mathematical patterns.
Knowing how to simplify correctly helps in solving problems faster and often significantly reduces the chances of mistakes. It makes subsequent calculations and operations easier to handle, especially when dealing with more complex equations.
Negative Exponents
Negative exponents can initially be confusing, but they are straightforward once you understand the concept. A negative exponent symbolizes division, which in turn represents the reciprocal of the base raised to the corresponding positive exponent.
  • Understanding: \(x^{-n} = \frac{1}{x^n}\) shows that negative exponents flip the base into a denominator.
  • Simplifying: Often in simplification, adding negative and positive exponents will eliminate the negative exponent, as seen with \(-6 + 12 = 6\).
  • Real-World Context: Negative exponents appear often in scientific notation, representing very small values across fields like physics and chemistry.
Gaining a strong foundation of negative exponents allows learners to navigate easily through complex calculations, ensuring clarity and accuracy. By converting negative exponents appropriately, you simplify expressions while preserving their value.

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