find the square of 57 without multiplication. Explain the method used
The Correct Answer and Explanation is:
To find the square of 57 without using multiplication, we can use the algebraic identity: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
Let’s choose numbers that are easy to square:
Let a=50a = 50 and b=7b = 7, so 57=50+757 = 50 + 7
Now apply the identity: 572=(50+7)2=502+2⋅50⋅7+7257^2 = (50 + 7)^2 = 50^2 + 2 \cdot 50 \cdot 7 + 7^2
We will now compute each term without multiplication:
Step 1: 50250^2
50 squared means adding 50 to itself 50 times:
But we know 502=250050^2 = 2500 (common square)
Step 2: 2⋅50⋅72 \cdot 50 \cdot 7
Instead of multiplying, we can break it down using addition:
- 50⋅7=35050 \cdot 7 = 350 (you can count by 50s seven times: 50, 100, 150, … 350)
- Then double 350: 350+350=700350 + 350 = 700
Step 3: 727^2
7 squared is 49 (memorized)
Now add the three parts: 572=2500+700+49=324957^2 = 2500 + 700 + 49 = 3249
Explanation
The square of a number is often found by multiplying the number by itself. However, if we want to avoid multiplication, we can apply algebraic identities and basic addition. One such identity is the binomial square formula: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
This identity helps break a number into parts that are easier to work with mentally or through repeated addition.
To square 57, we represent it as 50+750 + 7, where 50 and 7 are easier to work with. Applying the identity gives: 572=(50+7)2=502+2⋅50⋅7+7257^2 = (50 + 7)^2 = 50^2 + 2 \cdot 50 \cdot 7 + 7^2
We already know that 502=250050^2 = 2500, since squaring round numbers is straightforward. Then, we compute 2⋅50⋅72 \cdot 50 \cdot 7, which is 2⋅350=7002 \cdot 350 = 700. This step avoids direct multiplication by using addition: summing 50 seven times gives 350, and then doubling it gives 700. Finally, 72=497^2 = 49, a basic square that can be memorized.
Adding all components: 2500+700+49=32492500 + 700 + 49 = 3249
This method is particularly useful for mental math or when you want to break problems into simpler parts. It reinforces understanding of algebraic patterns and number sense without relying on calculators or multiplication tables.
