Find the squares of the following using $$ (a \pm b)^{2}=a^{2} \pm 2 a b+b^{2} $$ $$ \begin{array}{lllll} ext { (a) } 49 & ext { (b) } 81 & ext { (c) } 99 & ext { (d) } 1003 & ext { (e) } 999\end{array} $$

Question

Find the squares of the following using $$ (a \pm b)^{2}=a^{2} \pm 2 a b+b^{2} $$ $$ \begin{array}{lllll}	ext { (a) } 49 & 	ext { (b) } 81 & 	ext { (c) } 99 & 	ext { (d) } 1003 & 	ext { (e) } 999\end{array} $$

UpStudy AI · Accepted Answer

To find the squares of the given numbers using the formula $$ (a \pm b)^{2} = a^{2} \pm 2ab + b^{2} $$, we can express each number as a sum or difference of two numbers.

Let's calculate each square:

### (a) $$ 49^2 $$
We can express $$ 49 $$ as $$ (50 - 1) $$:
$$
49^2 = (50 - 1)^2 = 50^2 - 2 \cdot 50 \cdot 1 + 1^2
$$
Calculating each term:
$$
50^2 = 2500, \quad -2 \cdot 50 \cdot 1 = -100, \quad 1^2 = 1
$$
Putting it all together:
$$
49^2 = 2500 - 100 + 1 = 2401
$$

### (b) $$ 81^2 $$
We can express $$ 81 $$ as $$ (80 + 1) $$:
$$
81^2 = (80 + 1)^2 = 80^2 + 2 \cdot 80 \cdot 1 + 1^2
$$
Calculating each term:
$$
80^2 = 6400, \quad 2 \cdot 80 \cdot 1 = 160, \quad 1^2 = 1
$$
Putting it all together:
$$
81^2 = 6400 + 160 + 1 = 6561
$$

### (c) $$ 99^2 $$
We can express $$ 99 $$ as $$ (100 - 1) $$:
$$
99^2 = (100 - 1)^2 = 100^2 - 2 \cdot 100 \cdot 1 + 1^2
$$
Calculating each term:
$$
100^2 = 10000, \quad -2 \cdot 100 \cdot 1 = -200, \quad 1^2 = 1
$$
Putting it all together:
$$
99^2 = 10000 - 200 + 1 = 9801
$$

### (d) $$ 1003^2 $$
We can express $$ 1003 $$ as $$ (1000 + 3) $$:
$$
1003^2 = (1000 + 3)^2 = 1000^2 + 2 \cdot 1000 \cdot 3 + 3^2
$$
Calculating each term:
$$
1000^2 = 1000000, \quad 2 \cdot 1000 \cdot 3 = 6000, \quad 3^2 = 9
$$
Putting it all together:
$$
1003^2 = 1000000 + 6000 + 9 = 1006009
$$

### (e) $$ 999^2 $$
We can express $$ 999 $$ as $$ (1000 - 1) $$:
$$
999^2 = (1000 - 1)^2 = 1000^2 - 2 \cdot 1000 \cdot 1 + 1^2
$$
Calculating each term:
$$
1000^2 = 1000000, \quad -2 \cdot 1000 \cdot 1 = -2000, \quad 1^2 = 1
$$
Putting it all together:
$$
999^2 = 1000000 - 2000 + 1 = 998001
$$

### Summary of Results
- $$ 49^2 = 2401 $$
- $$ 81^2 = 6561 $$
- $$ 99^2 = 9801 $$
- $$ 1003^2 = 1006009 $$
- $$ 999^2 = 998001 $$
- $$ 49^2 = 2401 $$ - $$ 81^2 = 6561 $$ - $$ 99^2 = 9801 $$ - $$ 1003^2 = 1006009 $$ - $$ 999^2 = 998001 $$

Find the squares of the following using

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