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Chapter 6: Problem 384

Simplify. $$\frac{m^{12}}{\left(m^{8}\right)^{3}}$$

Short Answer

Expert verified
\[ \frac{1}{m^{12}} \].

Step by step solution

01

Understand the given expression

The given expression is \ \[ \frac{m^{12}}{(m^8)^3} \]. We need to simplify this by applying the properties of exponents.
02

Apply the power rule

According to the power rule, \ \[ (a^m)^n = a^{mn} \]. Apply this rule to the denominator: \ \[ (m^8)^3 = m^{8 \times 3} = m^{24} \]. The expression becomes \ \[ \frac{m^{12}}{m^{24}} \].
03

Apply the quotient rule

According to the quotient rule, \ \[ \frac{a^m}{a^n} = a^{m-n} \]. Apply this rule to the simplified expression from Step 2: \ \[ \frac{m^{12}}{m^{24}} = m^{12-24} = m^{-12} \].
04

Interpret the negative exponent

A negative exponent \ \[ a^{-n} \] can be rewritten as \ \[ \frac{1}{a^n} \].So, \ \[ m^{-12} = \frac{1}{m^{12}} \]. Therefore, the simplified expression is \ \frac{1}{m^{12}} \.

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Key Concepts

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

Power Rule
The power rule is a fundamental principle when working with exponents. It states that when you have an exponent raised to another exponent, you multiply the exponents together. For example, \( (a^m)^n = a^{mn} \).
This rule helps us simplify expressions by reducing the number of steps needed.
In the given exercise, we applied the power rule to the denominator \( (m^8)^3 \). This means multiplying the exponents 8 and 3 together:
\[(m^8)^3 = m^{8 \times 3} = m^{24}\]
Now, the expression is \ \frac{m^{12}}{m^{24}} \ which is simpler to work with.
Remember, multiplying exponents can make complex expressions easier to manage!
Quotient Rule
The quotient rule is another key concept for simplifying expressions with exponents. It tells us how to divide expressions with the same base. According to the quotient rule, when dividing like bases, you subtract the exponent of the denominator from the exponent of the numerator. This can be written as:
\[ \frac{a^m}{a^n} = a^{m-n} \]
In our exercise, after simplifying the denominator using the power rule, we applied the quotient rule:
\[(\frac{m^{12}}{m^{24}} = m^{12-24} = m^{-12})\]
This rule boils down a division problem into a simple subtraction, making the overall problem less daunting.
Negative Exponent
A negative exponent indicates that the base is on the wrong side of a fraction line, so to speak. It can be simplified by taking the reciprocal of the base and making the exponent positive. The rule looks like this:
\(a^{-n} = \frac{1}{a^n}\)
In the given problem, we ended up with a negative exponent after applying the quotient rule:
\[m^{-12} = \frac{1}{m^{12}}\]
This simplification shows how negative exponents create a fraction, which often simplifies the original expression. Understanding this can make handling exponents far less intimidating, turning what looks complex into a more manageable form.

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