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Chapter 6: Problem 385

Simplify. $$\frac{n^{8}}{\left(n^{6}\right)^{4}}$$

Short Answer

Expert verified
\( \frac{n^8}{(n^6)^4} = \frac{1}{n^{16}} \)

Step by step solution

01

Simplify the Denominator

Simplify the expression in the denominator \((n^6)^4 \). Use the power of a power rule \((a^m)^n = a^{m \times n}\). Therefore, \(n^6)^4 = n^{6 \times 4} = n^{24}\).
02

Write the Fraction

Rewrite the given fraction with the simplified denominator \ \(\frac{n^8}{n^{24}}\).
03

Apply the Quotient Rule

Simplify the fraction using the quotient rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\). Therefore, \(\frac{n^8}{n^{24}} = n^{8-24} = n^{-16}\).
04

Express in Positive Exponents

Express \(n^{-16}\) using positive exponents: \(n^{-16} = \frac{1}{n^{16}}\).

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Key Concepts

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

Power of a Power Rule
A fundamental concept when working with exponents is the power of a power rule.
When you see an expression like \( (a^m)^n \), you can simplify it using the rule \( (a^m)^n = a^{m \times n} \).
This means that you multiply the exponents.
For example, if you have \( (b^2)^3 \), this simplifies to \( b^{2 \times 3} = b^6 \).
In the given exercise, we apply this rule to \( (n^6)^4 \). By multiplying the exponents, we get \( n^{6 \times 4} = n^{24} \).
This step makes handling the original fraction easier.
Quotient Rule of Exponents
Another key concept is the quotient rule of exponents.
When you divide expressions with the same base, use the rule \( \frac{a^m}{a^n} = a^{m - n} \).
This rule tells us to subtract the exponent in the denominator from the exponent in the numerator.
In our example, after simplifying the denominator, we have \( \frac{n^8}{n^{24}} \).
Applying the quotient rule here: \( n^{8-24} = n^{-16} \).
It's a straightforward method to simplify fractions with exponents.
Positive Exponents
Finally, always try to express exponents as positive values.
A negative exponent essentially means the reciprocal.
In our case, \( n^{-16} \) can be written as \frac{1}{n^{16}} \, converting the negative exponent into a positive one.
This step is crucial because positive exponents often make it easier to interpret and understand the final result.
Simplifying expressions using these rules helps in getting clean and clear results.

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Most popular questions from this chapter

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