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Chapter 12: Problem 115

Simplify. $$ \frac{x^{12}}{x^{4}} $$

Short Answer

Expert verified
The simplified expression is \(x^{8}\).

Step by step solution

01

Identify the Division of Powers

The given expression is \(\frac{x^{12}}{x^{4}}\). We need to simplify this by using the laws of exponents.
02

Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, you have the same base \(x\) with exponents 12 and 4.
03

Subtract the Exponents

Subtract the exponents in the numerator and denominator: \(12 - 4 = 8\).
04

Write the Simplified Expression

Apply the result from the previous step to the base \(x\): \(x^{12-4} = x^{8}\).

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Key Concepts

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

Division of Powers
In this exercise, we simplify an expression involving division of powers. When you have similar bases, you can divide them by subtracting their exponents. For instance, consider the expression:
\ \ \( \frac{x^{12}}{x^{4}} \ \).
Here, we have the same base, which is \( x \), raised to different powers: \( 12 \) and \( 4 \).
Think of it like you are dividing repeated multiplications of the base. When dividing, you essentially cancel out the common factors.
This brings us to the laws of exponents, which guide us on how to handle such operations.
Laws of Exponents
Exponents are a shorthand way to express repeated multiplications of the same number or variable. The laws of exponents are crucial as they provide rules to perform exponent-related arithmetic easily.
Here are some useful laws for simplification:
  • Product Rule: \(a^m \cdot a^n = a^{m+n}\). Multiply powers with the same base by adding their exponents.
  • Power of a Power: \( (a^m)^n = a^{mn} \). Raise a power to another power by multiplying the exponents.
  • Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \). Divide powers with the same base by subtracting the exponents.
  • Zero Exponent Rule: \( a^0 = 1 \) (where \( a \) is not zero). Any non-zero number to the power of zero is 1.
Understanding these laws helps simplify and solve exponent-related problems efficiently. They act like a toolkit to tackle different scenarios involving exponents.
Quotient Rule for Exponents
The quotient rule for exponents is particularly helpful in the exercise given. This rule states:
\( \frac{a^m}{a^n} = a^{m-n} \).
You apply this rule when you divide powers of the same base. Here's how it works in our exercise:
Given: \( \frac{x^{12}}{x^{4}} \)
  • Identify the exponent in the numerator, which is \( 12 \).
  • Identify the exponent in the denominator, which is \( 4 \).
  • Subtract the exponent in the denominator from the exponent in the numerator: \( 12 - 4 = 8 \).
Now, rewrite the expression using the result: \( x^{12-4} = x^{8} \). This simplifies \( \frac{x^{12}}{x^{4}} \) to \( x^8 \).
By understanding and applying the quotient rule for exponents, you streamline algebraic simplifications involving powers.

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