Mathematics 10 — Exercise 5.3 · Q1 (i)

Full question

Simplify the following product or quotient of algebraic fractions, retaining the restrictions on the original denominators and divisors.

Step-by-step solution

1. Fraction 1: its original denominator cannot be zero.

   $$5\times l+10\times m\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$5\times l+10\times m=5\times l+10\times m$$

3. Take the common factor out of every term. Show each remaining term after division.

   $$5\times l+10\times m=5\times \left(l+2\times m\right)$$

4. Apply the matching factor identity: difference of squares, sum/difference of cubes, or grouping. Expand the displayed factors to check the identity.

   $$5\times l+10\times m=5\times \left(l+2\times m\right)$$

5. The complete factored form is:

   $$5\times \left(l+2\times m\right)$$

6. Fraction 2: its original denominator cannot be zero.

   $$16\times m\times n\ne 0$$

7. Expand any powers or inner brackets before factoring.

   $$5\times l^2-20\times m^2=5\times l^2-20\times m^2$$

8. Take the common factor out of every term. Show each remaining term after division.

   $$5\times l^2-20\times m^2=5\times \left(l^2-4\times m^2\right)$$

9. Apply the matching factor identity: difference of squares, sum/difference of cubes, or grouping. Expand the displayed factors to check the identity.

   $$5\times l^2-20\times m^2=5\times \left(l-2\times m\right)\times \left(l+2\times m\right)$$

10. The complete factored form is:

    $$5\times \left(l-2\times m\right)\times \left(l+2\times m\right)$$

11. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(24\times l\times m\right)\times \left(5\times \left(l-2\times m\right)\times \left(l+2\times m\right)\right)}{\left(16\times m\times n\right)\times \left(5\times \left(l+2\times m\right)\right)}$$

12. The original denominator must be nonzero. Keep this restriction even if a factor is later cancelled.

    $$\left(16\times m\times n\right)\times \left(5\times \left(l+2\times m\right)\right)\ne 0$$

13. Factor the numerator.

14. Expand any powers or inner brackets before factoring.

    $$\left(24\times l\times m\right)\times \left(5\times \left(l-2\times m\right)\times \left(l+2\times m\right)\right)=120\times l^3\times m-480\times l\times m^3$$

15. Take the common factor out of every term. Show each remaining term after division.

    $$120\times l^3\times m-480\times l\times m^3=\left(120\times l\times m\right)\times \left(l^2-4\times m^2\right)$$

16. Apply the matching factor identity: difference of squares, sum/difference of cubes, or grouping. Expand the displayed factors to check the identity.

    $$\left(24\times l\times m\right)\times \left(5\times \left(l-2\times m\right)\times \left(l+2\times m\right)\right)=120\times l\times m\times \left(l-2\times m\right)\times \left(l+2\times m\right)$$

17. The complete factored form is:

    $$120\times l\times m\times \left(l-2\times m\right)\times \left(l+2\times m\right)$$

18. Factor the denominator.

19. Expand any powers or inner brackets before factoring.

    $$\left(16\times m\times n\right)\times \left(5\times \left(l+2\times m\right)\right)=80\times l\times m\times n+160\times m^2\times n$$

20. Take the common factor out of every term. Show each remaining term after division.

    $$80\times l\times m\times n+160\times m^2\times n=\left(80\times m\times n\right)\times \left(2\times m+l\right)$$

21. Apply the matching factor identity: difference of squares, sum/difference of cubes, or grouping. Expand the displayed factors to check the identity.

    $$\left(16\times m\times n\right)\times \left(5\times \left(l+2\times m\right)\right)=80\times m\times n\times \left(l+2\times m\right)$$

22. The complete factored form is:

    $$80\times m\times n\times \left(l+2\times m\right)$$

23. Write the factored numerator over the factored denominator.

    $$\frac{120\times l\times m\times \left(l-2\times m\right)\times \left(l+2\times m\right)}{80\times m\times n\times \left(l+2\times m\right)}$$

24. This is the common factor of numerator and denominator.

    $$40\times l\times m+80\times m^2$$

25. Divide numerator and denominator by that same nonzero factor.

    $$\frac{\frac{\left(24\times l\times m\right)\times \left(5\times \left(l-2\times m\right)\times \left(l+2\times m\right)\right)}{40\times l\times m+80\times m^2}}{\frac{\left(16\times m\times n\right)\times \left(5\times \left(l+2\times m\right)\right)}{40\times l\times m+80\times m^2}}$$

26. Simplify the two divisions separately.

    $$\frac{3\times l^2-6\times l\times m}{2\times n}$$

27. Write the fraction in lowest form. Only factors, never added terms, have been cancelled.

    $$\frac{3\times l\times \left(l-2\times m\right)}{2\times n}$$

Answer

The final answer is shown below.