Squares & Cubes
Calculation League has two complex categories that relate exclusively to exponents:
Cubes/Squares category: exponents are limited to ^2 and ^3.
Exponents category: exponents range from ^2 through ^9.
In Calculation League Season 3, the playoff-stage “exponent” category was the most difficult category in the competition. Further modifications will be made to decrease the difficulty while still preserving the purpose of the category.
A later post will be made regarding the exponent category after further determinations regarding the revision of the category.
Example 1: 35^3 (easy level)
Our first example is simply cubing a two-digit number.
One often helpful approach for squaring a number is to recognize that:
(35 − 5) × (35 + 5) + 5^2 = 35^2
This is an instance of the identity:
(x − y)(x + y) + y^2 = x^2
In the case of a number that ends in 5, the application of this becomes extremely easy:
The square of all numbers ending in 5 ends in 25.
The preceding digits are a × (a + 1).
So:
35^2 = 3 × 4 = 12, append 25 → 1225
Now we multiply by 35 to cube:
35^3 = 35 × 35^2 = 1225 × 35
1225 × 35 = 1225 × (30 + 5) = 36750 + 6125 = 42875
Answer: 42875
Example 2: (8.7)^2 (easy level)
Now we have the square of a two-digit number, which is simply a 2×2 multiplication (after scaling).
Compute 87^2 first:
87^2 = (90 × 84) + 3^2 = 7560 + 9 = 7569
Since we are actually squaring 8.7, not 87, we divide by 10^2:
(8.7)^2 = 7569 / 100 = 75.69
So the answer is 75.69 (or 76 if rounded to the nearest integer).
Example 3: 354348^2 (advanced level)
Now we have the square of a six-digit number. Using algebraic shortcuts on the full number is going to be difficult. Most likely, the most effective approach is to treat this as a 6×6 multiplication, using your preferred multiplication strategy.
That said, the number has a nice structure: 354 and 348. We can take advantage of this by using three-digit blocks.
Step 1: Compute the key 3-digit squares/products
We start with a nearby reference:
350^2 = 122500 (as shown earlier)
We can build from 351^2:
351^2 = 350^2 + (2×350×1) + 1^2
351^2 = 122500 + 700 + 1 = 123201
Now adjust to 354 and 348.
For 354^2: compare 354 to 351 (difference +3)
(351 + 3)^2 = 351^2 + 2×351×3 + 3^2
2×351×3 = 2106, plus 9 → 2115
354^2 = 123201 + 2115 = 125316
For 348^2: compare 348 to 351 (difference −3)
(351 − 3)^2 = 351^2 − 2×351×3 + 3^2
351^2 − 2106 + 9 = 123201 − 2097 = 121104
For 348×354: note the symmetry around 351:
(351 − 3)(351 + 3) = 351^2 − 3^2 = 123201 − 9 = 123192
Step 2: Cross-multiply using three-digit blocks
Treat 354348 as 354|348.
Then:
354348^2 = (354×1000 + 348)^2
= (354^2)×10^6 + 2×(354×348)×10^3 + (348^2)
Plug in the computed pieces:
354^2 × 10^6 = 125316000000
2×(354×348)×10^3 = 2×123192×1000 = 246384000
348^2 = 121104
Now add:
125316000000
+000246384000
+000000121104
= 125562505104
Answer: 125562505104
Example 4: (93.5)^3 (advanced level)
Now we have the cube of a 3-digit number (after scaling).
A helpful first step is squaring 935 using the “ends in 5” shortcut:
935^2 → 93 × 94 with 25 appended
93 × 94 = 8742
so 935^2 = 874225
But we are working with 93.5, not 935, so:
(93.5)^2 = 874225 / 10^2 = 8742.25
Now we multiply to cube:
(93.5)^3 = 8742.25 × 93.5
A workable way to think about ×93.5 is:
×93 + ×0.5
Step 1: 8742 × 93
First do 87 × 93:
87 × 93 is 9 less than 90^2 (because 87×93 = (90−3)(90+3) = 8100 − 9)
so 87×93 = 8091
Then scale:
8700 × 93 = 809100
42 × 93 = 3906
809100 + 3906 = 813006
Step 2: Add the “half” and decimal adjustments
We need to add (8742/2), (93/4), and (1/8). This corresponds to the extra contributions coming from the .5 in 93.5 and the .25 in 8742.25.
Compute:
8742 / 2 = 4371
93 / 4 = 23.25
1 / 8 = 0.125
Sum: 4371 + 23.25 + 0.125 = 4394.375
Now total:
813006 + 4394.375 = 817400.375
In Calculation League, if we only need the nearest integer:
817400 is sufficient.
Example 5: (4559936)^2 (expert level)
This is the square of a seven-digit number. The default approach is to treat it as a 7×7 multiplication using cross multiplication or standard multiplication (if you are extremely accomplished).
But we can also use the Binomial Theorem for (x + y)^2 by picking a convenient nearby base.
Let:
x = 4,560,000
y = −64
Then:
(x + y)^2 = x^2 + 2xy + y^2
Step 1: x^2
x = 456 × 10^4, so:
x^2 = (456^2) × 10^8
Compute 456^2:
456^2 = (500×412) + 44^2
500×412 = 206000
44^2 = 1936
456^2 = 207936
So:
x^2 = 207936 × 10^8 = 20793600000000
Step 2: 2xy
2xy = 2×(4,560,000)×(−64) = −(2×64×4,560,000)
Note: 64 = 2^6, so 2×64 = 2^7 = 128.
So we want 456 × 128, then scale by 10^4:
Doubling 456 seven times:
456 → 912 → 1824 → 3648 → 7296 → 14592 → 29184 → 58368
So:
456×128 = 58368
therefore 4,560,000×128 = 58368×10^4 = 583680000
Because y is negative, this middle term is −583,680,000.
Step 3: y^2
y^2 = (−64)^2 = 64^2 = 4096
Combine terms
So:
(4,559,936)^2 = 20793600000000 − 583680000 + 4096
= 20793600000000 − 583675904
= 20793016324096
Answer: 20793016324096
To execute this quickly, you need to be comfortable chunking the subtraction—e.g., writing the leading digits from x^2, then subtracting the 9-digit middle adjustment cleanly.
Example 6: 17.7^2 (expert level)
On this last example, the question generation was unusually friendly. There are many ways to square a three-digit number (or do a 3×3 multiplication). Here are several approaches that all reach the same square:
Method A: Symmetric product + add-back
(177 − 77)(177 + 77) + 77^2
= 100×254 + 5929
= 25400 + 5929 = 31329
Method B: Another symmetric choice
(177 + 23)(177 − 23) + 23^2
= 200×154 + 529
= 30800 + 529 = 31329
Method C: Binomial Theorem
(170 + 7)^2 = 170^2 + 2×170×7 + 7^2
= 28900 + 2380 + 49 = 31329
Method D: Alternate binomial direction
(180 − 3)^2 = 180^2 − 2×180×3 + 3^2
= 32400 − 1080 + 9 = 31329
Method E: Factorization shortcut
177^2 = (59^2)(3^2) = 3481×9 = 31329
Now shift the decimal (since 17.7 = 177/10):
17.7^2 = 31329 / 100 = 313.29
Answer: 313.29