Exploring Algebraic Identities | Exercise 4.1

Question 2

Using the same identity, find the values of the following:

(i) (64)2(64)^2

(ii) (105)2(105)^2

(iii) (205)2(205)^2

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Solution
Understand the Question
  • To evaluate the square of a number close to a convenient base (like a multiple of 1010 or 100100) without long multiplication, express the number as a sum (a+b)(a + b).
  • Then, expand using the standard algebraic identity: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

(i) Using the same identity, find the value of (64)2(64)^2

Step 1 · Calculate (64)2(64)^2

Express 6464 as 60+460 + 4.Diagram 1

Using the identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 with a=60a = 60 and b=4b = 4:

(60+4)2=(60)2+2×60×4+(4)2=3600+480+16=4096\begin{aligned} (60 + 4)^2 &= (60)^2 + 2 \times 60 \times 4 + (4)^2 \\ &= 3600 + 480 + 16 \\ &= 4096 \end{aligned}
Answer

(i) 40964096

(ii) Using the same identity, find the value of (105)2(105)^2

Step 1 · Calculate (105)2(105)^2

Express 105105 as 100+5100 + 5.Diagram 2

Using the identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 with a=100a = 100 and b=5b = 5:

(100+5)2=(100)2+2×100×5+(5)2=10000+1000+25=11025\begin{aligned} (100 + 5)^2 &= (100)^2 + 2 \times 100 \times 5 + (5)^2 \\ &= 10000 + 1000 + 25 \\ &= 11025 \end{aligned}
Answer

(ii) 1102511025

(iii) Using the same identity, find the value of (205)2(205)^2

Step 1 · Calculate (205)2(205)^2

Express 205205 as 200+5200 + 5.Diagram 3

Using the identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 with a=200a = 200 and b=5b = 5:

(200+5)2=(200)2+2×200×5+(5)2=40000+2000+25=42025\begin{aligned} (200 + 5)^2 &= (200)^2 + 2 \times 200 \times 5 + (5)^2 \\ &= 40000 + 2000 + 25 \\ &= 42025 \end{aligned}
Answer

(iii) 4202542025

Common Mistakes
  • Forgetting the Middle Term: Writing (a+b)2=a2+b2(a + b)^2 = a^2 + b^2 instead of a2+2ab+b2a^2 + 2ab + b^2.
  • Zero Count in Squaring: Forgetting to double the number of zeros when squaring multiples of 1010 or 100100 (e.g., (200)2=40000(200)^2 = 40000, not 40004000).

More questions in Exercise 4.1

Q1

Using the identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, expand the following:

(i) (7x+4y)2(7x + 4y)^2

(ii) (75x+32y)2\left(\dfrac{7}{5}x + \dfrac{3}{2}y\right)^2

(iii) (2.5p+1.5q)2(2.5p + 1.5q)^2

(iv) (34s+8t)2\left(\dfrac{3}{4}s + 8t\right)^2

(v) (x+12y)2\left(x + \dfrac{1}{2y}\right)^2

(vi) (1x+1y)2\left(\dfrac{1}{x} + \dfrac{1}{y}\right)^2

Q2

Using the same identity, find the values of the following:

(i) (64)2(64)^2

(ii) (105)2(105)^2

(iii) (205)2(205)^2

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