Simplify (5/a-3/(a^2))/(3/(a^2)+5/a)
The given problem is a mathematical expression that involves simplifying a complex fraction or rational expression. The expression contains a combination of single fractions and compound fractions (fractions within fractions) involving variables. The task is to perform the simplification by applying the rules of algebra, particularly the properties of fractions, to combine and reduce the expression to its simplest form. This will likely involve finding a common denominator for the fractions in the numerator and the denominator, simplifying within each, and then dividing the simplified numerator by the simplified denominator.
$\frac{\frac{5}{a} - \frac{3}{a^{2}}}{\frac{3}{a^{2}} + \frac{5}{a}}$
Answer
Solution:
Step 1
Multiply both the top and bottom of the fraction by \(a^2\).
Step 1.1
Multiply \(\frac{5/a - 3/a^2}{3/a^2 + 5/a}\) by \(\frac{a^2}{a^2}\): \(\frac{a^2}{a^2} \cdot \frac{5/a - 3/a^2}{3/a^2 + 5/a}\)
Step 1.2
Combine terms: \(\frac{a^2(5/a - 3/a^2)}{a^2(3/a^2 + 5/a)}\)
Step 2
Distribute \(a^2\) across the terms in the numerator and denominator: \(\frac{a^2 \cdot 5/a - a^2 \cdot 3/a^2}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3
Simplify by canceling out common factors.
Step 3.1
Cancel the \(a\) terms.
Step 3.1.1
Extract \(a\) from \(a^2\): \(\frac{a \cdot a \cdot 5/a - a^2 \cdot 3/a^2}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.1.2
Cancel out the \(a\) terms: \(\frac{\cancel{a} \cdot a \cdot 5/\cancel{a} - a^2 \cdot 3/a^2}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.1.3
Rewrite the expression: \(\frac{a \cdot 5 - a^2 \cdot 3/a^2}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.2
Cancel the \(a^2\) terms.
Step 3.2.1
Move the negative sign in front of the fraction: \(\frac{a \cdot 5 - a^2 \cdot (-3)/a^2}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.2.2
Cancel out the \(a^2\) terms: \(\frac{a \cdot 5 - \cancel{a^2} \cdot (-3)/\cancel{a^2}}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.2.3
Rewrite the expression: \(\frac{a \cdot 5 + 3}{a^2 \cdot 3/a^2 + a^2 \cdot 5/a}\)
Step 3.3
Cancel the \(a^2\) terms.
Step 3.3.1
Cancel out the \(a^2\) terms: \(\frac{a \cdot 5 + 3}{\cancel{a^2} \cdot 3/\cancel{a^2} + a^2 \cdot 5/a}\)
Step 3.3.2
Rewrite the expression: \(\frac{a \cdot 5 + 3}{3 + a^2 \cdot 5/a}\)
Step 3.4
Cancel the \(a\) terms.
Step 3.4.1
Extract \(a\) from \(a^2\): \(\frac{a \cdot 5 + 3}{3 + a \cdot a \cdot 5/a}\)
Step 3.4.2
Cancel out the \(a\) terms: \(\frac{a \cdot 5 + 3}{3 + \cancel{a} \cdot a \cdot 5/\cancel{a}}\)
Step 3.4.3
Rewrite the expression: \(\frac{a \cdot 5 + 3}{3 + a \cdot 5}\)
Step 4
Rearrange the terms to place the constant in front of the variable: \(\frac{5a + 3}{3 + 5a}\)
Step 5
The expression is now simplified: \(\frac{5a + 3}{3 + 5a}\)
Knowledge Notes:
To simplify the given complex fraction, we apply several algebraic techniques:
Multiplication by the LCD: To eliminate the complex fraction, we multiply both the numerator and the denominator by the least common denominator (LCD), which is \(a^2\) in this case. This step gets rid of the fractions within the fraction.
Distributive Property: We distribute \(a^2\) across the terms in the numerator and the denominator to simplify the expression.
Cancellation: We cancel common factors in the numerator and the denominator to simplify the expression further. This involves recognizing that \(a^2/a = a\) and \(a^2/a^2 = 1\).
Rearrangement: We rearrange the terms to place the constant in front of the variable for a more conventional form.
Final Simplification: After canceling and rearranging, we arrive at the simplest form of the original expression.
Throughout the process, we must be careful to apply algebraic rules correctly and ensure that we do not cancel terms incorrectly or change the signs of the terms.
