Simplify \frac{\left(\frac{5}{\left(a+h\right)^2}-\frac{5}{a^2}\right)}{h}
To simplify, first find the common denominator for the two fractions in the numerator. The common denominator is \(a^2 \cdot (a+h)^2\).
To simplify the expression \frac{\left(\frac{5}{\left(a+h\right)^2}-\frac{5}{a^2}\right)}{h}, you can follow these steps:
Step 1: Start by finding a common denominator for the fractions within the parentheses. The common denominator in this case is a^2(a + h)^2.
Step 2: Rewrite each fraction with the common denominator:
\frac{5 \cdot a^2}{a^2 \cdot (a + h)^2} \text{ and } \frac{5 \cdot (a + h)^2}{a^2 \cdot (a + h)^2}
Step 3: Subtract the second fraction from the first fraction:
\frac{5 \cdot a^2}{a^2 \cdot (a + h)^2} - \frac{5 \cdot (a + h)^2}{a^2 \cdot (a + h)^2}
Step 4: Combine the fractions with the same denominator:
\frac{5a^2 - 5(a + h)^2}{a^2 \cdot (a + h)^2}
Step 5: Expand the numerator:
\frac{5a^2 - 5(a^2 + 2ah + h^2)}{a^2 \cdot (a + h)^2}
Step 6: Simplify the numerator:
\frac{5a^2 - 5a^2 - 10ah - 5h^2}{a^2 \cdot (a + h)^2}
Step 7: Combine like terms:
\frac{-10ah - 5h^2}{a^2 \cdot (a + h)^2}
Step 8: Factor out a common factor of -5h:
\frac{-5h(2a + h)}{a^2 \cdot (a + h)^2}
Thus, the simplified expression is \frac{-5h(2a + h)}{a^2 \cdot (a + h)^2}.
To simplify the given expression, let's break it down step by step:
Step 1: Start by finding a common denominator for the fractions on the numerator side. The common denominator should be the product of the denominators of both fractions, which in this case is \((a+h)^2 \cdot a^2\).
Step 2: Rewrite both fractions with the common denominator.
The first fraction becomes \(\frac{5 \cdot a^2}{(a+h)^2 \cdot a^2}\).
The second fraction becomes \(\frac{5 \cdot (a+h)^2}{(a+h)^2 \cdot a^2}\).
Step 3: Combine the fractions on the numerator side by subtracting them. Be sure to keep the common denominator.
\(\frac{5 \cdot a^2 - 5 \cdot (a+h)^2}{(a+h)^2 \cdot a^2}\)
Step 4: Expand the binomial \((a+h)^2\) by squaring it.
\(\frac{5 \cdot a^2 - 5 \cdot (a^2 + 2ah + h^2)}{(a+h)^2 \cdot a^2}\)
Step 5: Simplify the numerator by distributing the negative sign and multiplying.
\(\frac{5a^2 - 5a^2 - 10ah - 5h^2}{(a+h)^2 \cdot a^2}\)
Step 6: Combine like terms in the numerator.
\(\frac{-10ah - 5h^2}{(a+h)^2 \cdot a^2}\)
Step 7: Factor out a common factor (h) from the numerator.
\(\frac{-h(10a + 5h)}{(a+h)^2 \cdot a^2}\)
Step 8: Divide the whole expression by h to simplify.
\(\frac{-10a - 5h}{(a+h)^2 \cdot a^2}\)
So, the simplified form of the given expression is \(\frac{-10a - 5h}{(a+h)^2 \cdot a^2}\).