## Simplification Questions, Formulas, Shortcuts and Practice Problems

**Rule of Simplification**

- In simplifying an expression, first of all vinculum or bar must be removed. For example: we know that -4-5 = -9 but, -[latex ]\overline{4-5}[/latex] = -(- 1) = 1
- After removing the bar, the brackets must be removed, strictly in the order (), {} and [].
- After removing the brackets, we must use the following operations strictly in the order given below, (a) of (b) division (c) multiplication (d) addition and (e) subtraction

Ex 1:

Solution:

### Simplification Questions on Application of Algebra Formulas

2. **(a + b) ^{2} = a^{2} + b^{2} + 2ab**

Ex 2: Simplify 0.46 x 0.46 + 0.54 x 0.54 + 0.92 x 0.54

Solution:

We have the expression

0.46 x 0.46 + 0.54 x 0.54 + 0.92 x 0.54 = (0.46)^{2} + (0.54)^{2} + 2. x 0.46 x 0.54

If we suppose a = 0.46 and b = 0.54, then

= a^{2} + b^{2} + 2ab = (a + b)^{2}

= (0.46 + 0.54)^{2} = (1.00)^{2} = 1

∴ Answer = 1.

3. **(a – b) ^{2} = a^{2} +b^{2} – 2ab**

Ex 3. simplify 1.66 × 1.66 + 0.66 × 0.66 – 1.32 × 1.66

Solution:

We have the expression

1.66 × 1.66 + 0.66 × 0.66 – 1.32 × 1.66

Now, applying the above formula,

= (1.66 – 0.66)^{2} = (1)^{2} = 1

∴ Answer = 1

4. **(a + b) ^{2} +(a- b)^{2} = 2 (a^{2} + b^{2})**

**Ex 4: Simplify the following**

2[1.25 x 1.25 + 0.25 x 0.25]

Solution:

Applying the above formula, we have

2[(1.25)

^{2}+ (0.25)

^{2}] = (1.25 + 0.25)

^{2}+ (1.25 – 0.25)

^{2}

= (1.5)

^{2}+ (1)

^{2}

= 2.25 + 1 = 3.25

5. **(a + b) ^{2} – (a- b)^{2} = 4ab**

Ex 5:

Solution:

Applying the above formula, we have a = 14.5, and b = 6.23

6. **(a + b) (a – b) = a ^{2} –b^{2}**

Ex 6:

Simplify (50^{2 }– 40^{2}) =? X 45

Solution:

Suppose a = 50 and b = 40

And required number = x

Applying the above formula,

∴ Required answer = 20.

7. **(a + b) ^{3} = a^{3} + + 3a^{2}b + 3ab^{2} + b^{3 }= a^{3} + b^{3} + 3ab (a + b)**

Ex 7:

Simplify (0.6)^{3} + (0.4)^{3} + 3 x 0.6 x 0.4(0.6 + 0.4)

Solution:

The above expression can be written as

(0.6)^{3} + (0.4)^{3} + 3 x 0.6 x 0.4(0.6 + 0.4)

Now, we suppose 0.6 = a and 0.4 = b and applying the above formula, we have (0.6 + 0.4)^{3} = (1)^{3} = 1

8. **(a – b) ^{3} = a^{3} – 3a^{2}b + 3ab^{2} – b^{3} = a^{3} – b^{3} – 3ab (a-b)**

Ex 8:

Simplify

Solution:

9. **a ^{3} +b^{3} = (a + b)(a^{2} – ab + b^{2})**

Solution:

Here a = 0.5 and b = 0.4

∴ Required answer = 0.5 + 0.4 = 0.9

10. **a ^{3} – b^{3} = (a- b) (a^{2} +ab + b^{2})**

Ex 10:

Solution:

### Simplification Questions – Practice Problems

### Simplification Practice Problems Answers

1. 4505

2. 2

3. 49

4. 10

5. 104

6. 108.45

7. 0.46

8. 315

9. [latex s=2]13 \frac { 3 } { 10 }[/latex]

10. [latex s=2] \frac { 3 } { 7 }[/latex]

11. 12

12. 2

13. [latex s=2] \frac { 61 } { 11 }[/latex]

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