Finding product without working it out

Question

What's the easy way to find the solution for the below problem without actually multiplying the numbers:

$$(24 * 24) - (16 * 16)?$$

I tried multiplying the numbers but that is a long way.

The answer is in the below format

A) $(8 \times 8)$

B) $(40 \times 40)$

C) $(8 \times 2)$

D) $(8 \times 40)$

Answer 1

Note that this expression is of the form $a^2 - b^2$ for $a = 24$ and $b = 16$. This expression is a difference of two squares, so it has the factorisation $$a^2 - b^2 = (a - b)(a+b).$$ Using the values of $a$ and $b$ we have $$24^2 - 16^2 = (24 - 16)(24 + 16) = 8\times 40 = 320.$$

Answer 2

You can use the formula $a^2-b^2 = (a-b)(a+b)$. You can also observe that all numbers in the expression are divisible by 8 and factor an $8^2$ out to get $8^2(3*3-2*2)$.

Answer 3

You can also try and use prime factorization:

\begin{align}(24 \cdot 24) - (16 \cdot 16) &= 24^2 - 16^2 \\ &= (3\cdot 2^3)^2 - (2^4)^2 \\ &= (3^2 \cdot2^6) - (2^8) \\ &= 2^6 \bigg[3^2 - 2^2 \bigg] \\ &= 2^6 \bigg[9 - 4 \bigg] \\ &= 2^6 (5) \\ &= 64(5) \\ &= 320\end{align}