Express
\[
\frac{5 x}{x^{3}-5 x^{2}} \div \frac{x^{2}+x-20}{x^{2}-5 x}
\]
as a single fraction in its simplest form.
Input Note: Enter answer in fully factorised form.
Answer
Check the result. Substitute a value for \(x\) (for example, \(x = 1\)) into both the original expression and the simplified expression. They should give the same result. For the original expression: \(\frac{5 \times 1}{1^{3}-5 \times 1^{2}} \div \frac{1^{2}+1-20}{1^{2}-5 \times 1} = -1\). For the simplified expression: \(\frac{5}{1 \times (1-4) \times (1+5)} = -1\). Since both expressions give the same result, the simplified form is correct.
Step 1 ：Factorise the denominators and numerators where possible. The expression \(\frac{5 x}{x^{3}-5 x^{2}} \div \frac{x^{2}+x-20}{x^{2}-5 x}\) becomes \(\frac{5 x}{x^{2}(x-5)} \div \frac{(x-4)(x+5)}{x(x-5)}\).
Step 2 ：Division of fractions is the same as multiplication by the reciprocal. So, the expression becomes \(\frac{5 x}{x^{2}(x-5)} \times \frac{x(x-5)}{(x-4)(x+5)}\).
Step 3 ：Simplify the expression by cancelling out common factors. The expression becomes \(\frac{5 x}{x^{2}} \times \frac{1}{(x-4)(x+5)}\).
Step 4 ：Simplify further. The expression becomes \(\frac{5}{x} \times \frac{1}{(x-4)(x+5)}\).
Step 5 ：Multiply the fractions. The final simplified expression is \(\boxed{\frac{5}{x(x-4)(x+5)}}\).
Step 6 ：Check the result. Substitute a value for \(x\) (for example, \(x = 1\)) into both the original expression and the simplified expression. They should give the same result. For the original expression: \(\frac{5 \times 1}{1^{3}-5 \times 1^{2}} \div \frac{1^{2}+1-20}{1^{2}-5 \times 1} = -1\). For the simplified expression: \(\frac{5}{1 \times (1-4) \times (1+5)} = -1\). Since both expressions give the same result, the simplified form is correct.
