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Chapter 10: Problem 580

Simplify. $$\frac{\left(n^{5}\right)^{3}}{\left(n^{2}\right)^{8}}$$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{n}\).

Step by step solution

01

Apply the Power Rule

The Power Rule states that \(\big(a^m)^n = a^{m \times n}\). Apply this rule separately to both the numerator \(\big( n^5 \big)^3\) and the denominator \(\big( n^2 \big)^8\). Thus, the expression becomes \(\frac{n^{15}}{n^{16}}\).
02

Apply the Quotient Rule

The Quotient Rule states that \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to simplify \(\frac{n^{15}}{n^{16}}\). Subtract the exponents in the numerator and the denominator: \(\frac{n^{15}}{n^{16}} = n^{15-16} = n^{-1}\).
03

Simplify Further

Simplify \(\frac{1}{n}\), since \(\frac{1}{n^{1}}\) can be written as \(\frac{1}{n}\). Hence, \(\frac{\big(n^{5}\big)^{3}}{\big(n^{2}\big)^{8}} = \frac{1}{n}\).

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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 very handy when you have an exponent raised to another exponent. It states that \((a^m)^n = a^{m \times n}\). In simple terms, you multiply the exponents together. For example, if you have \((n^5)^3\), you use the Power Rule to get \((n^{5 \times 3} = n^{15})\). This makes it easier to handle complex expressions involving exponents.
Quotient Rule
The Quotient Rule helps simplify a fraction where both the numerator and the denominator have the same base but different exponents. According to this rule, \(\frac{a^m}{a^n} = a^{m-n}\). You subtract the exponent in the denominator from the exponent in the numerator. For instance, \(\frac{n^{15}}{n^{16}}\) can be simplified by subtracting the exponents: \((15 - 16)\), giving you \((n^{-1})\).
Negative Exponents
Negative exponents might look tricky, but they're straightforward to simplify. A negative exponent means you take the reciprocal of the base raised to the positive exponent. So \(a^{-n} = \frac{1}{a^n}\). In our example, \(n^{-1}\) turns into \(\frac{1}{n^1} = \frac{1}{n}\).
Fraction Simplification
Simplifying fractions involves reducing them to their simplest form. If your fraction contains variables with exponents, use exponent rules to simplify as much as possible. With our problem, \(\frac{(n^5)^3}{(n^2)^8}\), we end up with \(\frac{1}{n}\). Once you鈥檝e used the Power Rule and Quotient Rule, the fraction is already reduced to its simplest form. That鈥檚 simplifying at its best!

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