Question

Which of the following is the lowest?

• A. \(30^2 - 12^2 - 3^2\)
• B. \(35^2 - 14^2 - 6^2\)
• C. \(30^2 - 10^2 - 5^3\)
• D. \(33^2 - 17^2 - 1^2\)

Hint:

Work out each expression fully before comparing; option (3) has a cube in its last term, not a square, so do not skip that detail.

Answer 1

The Correct Option is C

Step 1: Use the difference of squares for the first two terms.
For each option, the first subtraction has the form \(x^2 - y^2 = (x-y)(x+y)\), which is often quicker than squaring two large numbers separately.

Step 2: Apply this to option (1).
\[ 30^2 - 12^2 = (30-12)(30+12) = 18 \times 42 = 756 \]
Then subtract the last term: \(756 - 3^2 = 756 - 9 = 747\).

Step 3: Apply this to option (2).
\[ 35^2 - 14^2 = (35-14)(35+14) = 21 \times 49 = 1029 \]
Then subtract the last term: \(1029 - 6^2 = 1029 - 36 = 993\).

Step 4: Apply this to option (3), keeping the cube intact.
\[ 30^2 - 10^2 = (30-10)(30+10) = 20 \times 40 = 800 \]
Then subtract \(5^3 = 125\): \(800 - 125 = 675\).

Step 5: Apply this to option (4).
\[ 33^2 - 17^2 = (33-17)(33+17) = 16 \times 50 = 800 \]
Then subtract the last term: \(800 - 1^2 = 800 - 1 = 799\).

Step 6: Compare.
The values are 747, 993, 675 and 799, in that order. The lowest is 675, from option (3).
\[ \boxed{675} \]