Simplifying the Rational Expression
To simplify the expression \( \frac{10n^2}{24n^4} + \frac{9n^2}{18} \), we first need to break down both fractions individually.
Step 1: Simplify Each Fraction
1. For the first fraction, \( \frac{10n^2}{24n^4} \):
- The numerator is 10 and the denominator is 24.
- Both numbers can be divided by 2, resulting in \( \frac{5}{12} \).
- The variable part is \( n^2 \) in the numerator and \( n^4 \) in the denominator.
- Applying the rule \( \frac{a^m}{a^n} = a^{m-n} \), we have \( \frac{n^2}{n^4} = n^{-2} \).
Thus, the first fraction simplifies to:
\[ \frac{10n^2}{24n^4} = \frac{5}{12} n^{-2} = \frac{5}{12n^2} \]
2. For the second fraction, \( \frac{9n^2}{18} \):
- The numerator is 9 and the denominator is 18.
- Dividing both by 9 gives \( \frac{1}{2} \).
Hence, the second fraction simplifies to:
\[ \frac{9n^2}{18} = \frac{1}{2} n^2 \]
Step 2: Combine the Simplified Fractions
Now we combine the simplified fractions:
\[ \frac{5}{12n^2} + \frac{1}{2} n^2 \]
Finding a Common Denominator
The common denominator between \( 12n^2 \) and \( 2 \) is \( 12n^2 \). Now we can convert the second term:
\[ \frac{1}{2} n^2 = \frac{6n^4}{12n^2} \]
Combine Like Terms
Now we can rewrite the expression as:
\[ \frac{5}{12n^2} + \frac{6n^4}{12n^2} = \frac{5 + 6n^4}{12n^2} \]
Restrictions on the Variable
In simplifying rational expressions, we need to consider the values that would lead to division by zero. Here, the expression \( \frac{5 + 6n^4}{12n^2} \) has a denominator of \( 12n^2 \). This means we cannot have:
- \( n^2 = 0 \) which implies \( n = 0 \).
Therefore, the restriction on the variable is:
\[ n \neq 0 \]
Final Result
The simplified form of the rational expression is:
\[ \frac{5 + 6n^4}{12n^2} \]
with the restriction that:
\[ n \neq 0 \]