Problem 50 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

1

Step 1 - 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: \[ \frac{9 n^3}{54 n^8} = \frac{n^3}{6 n^8} \]

Step 2 - 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 \[ \frac{n^{-5}}{6} \]

Step 3 - Apply the Power of Zero

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

Key Concepts

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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