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Chapter 5: Problem 50

\(\left(\frac{9 n^{3}}{54 n^{8}}\right)^{0}\)

Short Answer

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1

Step by step solution

01

- Simplify the Fraction

Simplify the fraction inside the parentheses first. The fraction is \(\frac{9 n^{3}}{54 n^{8}}\). Divide the numerator and denominator by 9: \[ \frac{9 n^3}{54 n^8} = \frac{n^3}{6 n^8} \]
02

- Simplify the Power

Simplify \(\frac{n^3}{6 n^8}\) further by using the rule of exponents which states \( \frac{a^m}{a^n} = a^{m-n} \). Here, \(\frac{n^3}{n^8} = n^{3-8} = n^{-5}\), and thus the fraction becomes \[ \frac{n^{-5}}{6} \]
03

- Apply the Power of Zero

Recall that any non-zero number raised to the power of zero equals 1. In this case, \(\big(\frac{n^{-5}}{6}\big)^{0} = 1 \).

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Key Concepts

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

Fractions
Fractions represent a part of a whole. They consist of a numerator (top number) and a denominator (bottom number). For example, in the fraction \(\frac{9}{54}\), 9 is the numerator, and 54 is the denominator.
In algebraic fractions, we can also have variables. For instance, \(\frac{9n^3}{54n^8}\) has both numbers and the variable \(n\).
To simplify such fractions, we divide both the numerator and denominator by their greatest common factor (GCF).
In the provided exercise, dividing both by 9 gives us \(\frac{n^3}{6n^8}\). Remember, simplifying fractions makes further calculations easier!
Exponents
Exponents represent repeated multiplication. For example, \(n^3\) means \(n \times n \times n\). When simplifying expressions with exponents, it's important to remember some key rules:
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