Mathematics 10 — Exercise 5.3 · Q1 (vi)

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.

   $${\left(a^2-a\times b\right)}^2\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $${\left(a^2+a\times b\right)}^2=a^4+2\times a^3\times b+a^2\times b^2$$

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

   $$a^4+2\times a^3\times b+a^2\times b^2=a^2\times \left(a^2+2\times a\times b+b^2\right)$$

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

   $${\left(a^2+a\times b\right)}^2=a^2\times {\left(a+b\right)}^2$$

5. The complete factored form is:

   $$a^2\times {\left(a+b\right)}^2$$

6. Expand any powers or inner brackets before factoring.

   $${\left(a^2-a\times b\right)}^2=a^4-2\times a^3\times b+a^2\times b^2$$

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

   $$a^4-2\times a^3\times b+a^2\times b^2=a^2\times \left(a^2-2\times a\times b+b^2\right)$$

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

   $${\left(a^2-a\times b\right)}^2=a^2\times {\left(a-b\right)}^2$$

9. The complete factored form is:

   $$a^2\times {\left(a-b\right)}^2$$

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

    $${\left(a-b\right)}^2\ne 0$$

11. A divisor fraction must also be nonzero, so its numerator cannot be zero.

    $${\left(a+b\right)}^2\ne 0$$

12. Change division by this fraction into multiplication by its reciprocal.

    $$\frac{{\left(a-b\right)}^2}{{\left(a+b\right)}^2}$$

13. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(a^2\times {\left(a+b\right)}^2\right)\times {\left(a-b\right)}^2}{\left(a^2\times {\left(a-b\right)}^2\right)\times {\left(a+b\right)}^2}$$

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

    $$\left(a^2\times {\left(a-b\right)}^2\right)\times {\left(a+b\right)}^2\ne 0$$

15. Factor the numerator.

16. Expand any powers or inner brackets before factoring.

    $$\left(a^2\times {\left(a+b\right)}^2\right)\times {\left(a-b\right)}^2=a^6-2\times a^4\times b^2+a^2\times b^4$$

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

    $$a^6-2\times a^4\times b^2+a^2\times b^4=a^2\times \left(a^4-2\times a^2\times b^2+b^4\right)$$

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

    $$\left(a^2\times {\left(a+b\right)}^2\right)\times {\left(a-b\right)}^2=a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2$$

19. The complete factored form is:

    $$a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2$$

20. Factor the denominator.

21. Expand any powers or inner brackets before factoring.

    $$\left(a^2\times {\left(a-b\right)}^2\right)\times {\left(a+b\right)}^2=a^6-2\times a^4\times b^2+a^2\times b^4$$

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

    $$a^6-2\times a^4\times b^2+a^2\times b^4=a^2\times \left(a^4-2\times a^2\times b^2+b^4\right)$$

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

    $$\left(a^2\times {\left(a-b\right)}^2\right)\times {\left(a+b\right)}^2=a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2$$

24. The complete factored form is:

    $$a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2$$

25. Write the factored numerator over the factored denominator.

    $$\frac{a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2}{a^2\times {\left(a-b\right)}^2\times {\left(a+b\right)}^2}$$

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

    $$a^6-2\times a^4\times b^2+a^2\times b^4$$

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

    $$\frac{\frac{\left(a^2\times {\left(a+b\right)}^2\right)\times {\left(a-b\right)}^2}{a^6-2\times a^4\times b^2+a^2\times b^4}}{\frac{\left(a^2\times {\left(a-b\right)}^2\right)\times {\left(a+b\right)}^2}{a^6-2\times a^4\times b^2+a^2\times b^4}}$$

28. Simplify the two divisions separately.

    $$\frac{1}{1}$$

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

    $$1$$

Answer

The final answer is shown below.