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Chapter 4: Problem 57

\(\frac{x^{12}}{x^{-3}}\)

Short Answer

Expert verified
The expression simplifies to \( x^{15} \).

Step by step solution

01

Simplify the division of exponents

Recall the property of exponents that states \ \ \frac{a^m}{a^n} = a^{m-n} \ \ In this exercise, apply it to the given expression \ \ \frac{x^{12}}{x^{-3}} \ \ Using the property, the expression simplifies to \( x^{12 - (-3)} \)
02

Simplify the exponent

Calculate the exponent by simplifying the expression inside the exponent: \ \ 12 - (-3) = 12 + 3 = 15. \ \ This gives us \( x^{15} \)
03

Final Answer

Thus, the simplified form of the given expression is \( x^{15} \)

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is crucial for simplifying expressions like \(\frac{x^{12}}{x^{-3}}\). These properties help us manipulate and simplify expressions with ease. Here are some key properties to remember:

\begin{itemize}
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

    • The Quotient of Powers property is particularly useful for the given problem. It tells us that when we divide two exponents with the same base, we subtract the exponent of the denominator from the exponent of the numerator. This makes simplifying complex expressions much easier.

    Division of Exponents
    Division of exponents relies on the property that allows us to subtract the exponents when the base is the same. For example, in the given problem \(\frac{x^{12}}{x^{-3}}\), we can simplify this division by subtracting the exponent in the denominator from the exponent in the numerator.

    \begin{itemize}
  • First, recall the property: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Apply this property to our problem: \(\frac{x^{12}}{x^{-3}} = x^{12 - (-3)}\)

    • Notice that we need to subtract \-3\, which is the same as adding 3. This simplifies our expression to \x^{12 + 3}\.

    Simplification Steps
    The simplification process involves just a few straightforward steps for solving \(\frac{x^{12}}{x^{-3}}\). Let鈥檚 break it down:

    \begin{itemize}
  • Step 1: Use the Quotient of Powers property: \(\frac{a^m}{a^n} = a^{m-n}\)
    ewline We apply this to our problem: \(\frac{x^{12}}{x^{-3}} = x^{12 - (-3)}\)
  • Step 2: Simplify the exponent: \(12 - (-3)\) is the same as \(12 + 3\), which equals \(15\).
    ewline This gives us \(x^{15}\)
  • Step 3: Write the final answer: The simplified form of our given expression is \(x^{15}\).

    • By understanding and applying these steps, we can simplify expressions involving exponents quickly and accurately.

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