# Using identities, evaluate

i) 71^{2 }ii) 99^{2 }iii) 102^{2 }iv) 998^{2 }v) 5.2^{2}

vi) 297 × 303 vii) 78 × 82 viii) 8.9^{2 }ix) 10.5 × 9.5

**Solution:**

The three basic algebraic identities, which we will be using to evaluate the expressions are as follows.

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

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

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

(i) 71^{2}

= (70 + 1)^{2}

= (70)^{2} + 2(70)(1) + (1)^{2} [Since, (a + b)^{2} = a^{2} + 2ab + b^{2}]

= 4900 + 140 + 1

= 5041

(ii) 99^{2}

= (100 - 1)^{2}

= (100)^{2} - 2(100)(1) + (1)^{2} [Since, (a - b)^{2} = a^{2} - 2ab + b^{2}]

= 10000 - 200 + 1

= 9801

(iii) 102^{2}

= (100 + 2)^{2}

= (100)^{2} + 2(100)(2) + (2)^{2} [Since, (a + b)^{2} = a^{2} + 2ab + b^{2}]

= 10000 + 400 + 4

= 10404

(iv) 998^{2}

= (1000 - 2)^{2}

= (1000)^{2} - 2(1000)(2) + (2)^{2} [Since, (a - b)^{2} = a^{2} - 2ab + b^{2}]

= 1000000 - 4000 + 4

= 996004

(v) (5.2)^{2}

= (5.0 + 0.2)^{2}

= (5.0)^{2} + 2(5.0)(0.2) + (0.2)^{2} [Since, (a + b)^{2} = a^{2} + 2ab + b^{2}]

= 25 + 2 + 0.04

= 27.04

(vi) 297 × 303

= (300 - 3) × (300 + 3) [Since, (a + b)(a - b) = a^{2} - b^{2}]

= (300)^{2} - (3)^{2}

= 90000 - 9

= 89991

(vii) 78 × 82

= (80 - 2) × (80 + 2) [Since, (a + b)(a - b) = a^{2} - b^{2}]

= (80)^{2} - (2)^{2}

= 6400 - 4

= 6396

(viii) 8.9^{2}

= (9.0 - 0.1)^{2}

= (9.0)^{2} - 2(9.0)(0.1) + (0.1)^{2} [Since, (a - b)^{2} = a^{2} - 2ab + b^{2}]

= 81 - 1.8 + 0.01

= 79.21

(ix) 10.5 × 9.5

= (10 + 0.5) × (10 - 0.5)

= 10^{2} - 0.5^{2} [Since, (a + b)(a - b) = a^{2} - b^{2}]

= 100 - 0.25

= 99.75

**☛ Check: **NCERT Solutions for Class 8 Maths Chapter 9

**Video Solution:**

## Using identities, evaluate. i) 71²^{ }ii) 99²^{ }iii) 102²^{ }iv) 998²^{ }v) (5.2)²^{ }vi) 297 × 303 vii) 78 × 82 viii) 8.9²^{ }ix) 10.5 × 9.5

NCERT Solutions Class 8 Maths Chapter 9 Exercise 9.5 Question 6

**Summary:**

Using identities, the following expressions i) 71^{2 }ii) 99^{2 }iii) 102^{2 }iv) 998^{2 }v) (5.2)^{2 }vi) 297 × 303 vii) 78 × 82 viii) 8.9^{2 }ix) 10.5 × 9.5 are evaluated as follows i) 5041 ii) 9801 iii) 10404 iv) 996004 v) 27.04 vi) 89991 vii) 6396 viii) 79.21 ix) 99.75

**☛ Related Questions:**

- Use a suitable identity to get each of the following products. (i) (x + 3)(x + 3) (ii) (2y + 5)(2y + 5) (iii) (2a - 7)(2a - 7) (iv) (3a - (1/2))(3a - (1/2)) (v) (1.1 m - 0.4)(1.1 m + 0.4) (vi)(a2 + b2)(-a2 + b2) (vii) (6x - 7)(6x + 7) (viii) (-a + c)(-a + c) (ix) (x/2 + 3y/4)(x/2 + 3y/4) (x) (7a - 9b)(7a - 9b)
- Use the identity (x + a)(x + b) = x2 + (a + b)x + ab to find the following products. (i) (x + 3)(x + 7) (ii) (4x + 5)(4x + 1) (iii) (4x - 5)(4x -1) (iv) (4x + 5)(4x -1) (v) (2x + 5y)(2x + 3 y) (vi) (2a2 + 9)(2a2 + 5) (vii) (xyz - 4)(xyz - 2)
- Find the following squares by using the identities. (i)(b - 7)2 (ii) (xy + 3z)2 (iii) (6x2 - 5 y)2 (iv) (2m/3 + 3n/2)2 (v) (0.4 p - 0.5q)2 (vi) (2xy + 5 y)2
- Simplify (i)(a2 - b2)2 (ii) (2x + 5)2 - (2x - 5)2 (iii) (7m - 8n)2 + (7m + 8n)2 (iv) (4m + 5n)2 + (5m + 4n)2 (v) (2.5p - 1.5q)2 - (1.5p - 2.5q)2 (vi) (ab + bc)2 - 2ab2c (vii) (m2 - n2m)2 + 2m3n2

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