Evaluating 973 Using Binomial Cube Formula and Mental Math Techniques

When evaluating 973, we can use the binomial cube formula or mental math to find the result more efficiently. Let's explore both methods step-by-step.

Using the Binomial Cube Formula

The expanded form of the binomial cube is given by:

x3?3x2y 3xy2?y3

In this context, let a be 100 and b be 3. Applying the formula, we get:

a3?3a2b 3ab2?b3

Substituting a and b with 100 and 3, respectively, we simplify:

100?331003?310023100?33

This simplifies to:

1000000?27?900002700912673

Thus, we determine that 973 is equal to 912673.

Mental Math Techniques

Another effective method is to break down the problem using mental math. We start by utilizing the identity:

a?b3a3?3a2b 3ab2?b3

With a 100 and b 3, we proceed as follows:

9731003?33

We know that 1003 is 1000000 and 33 is 27. Therefore:

1000000?27999973

Next, we subtract the remaining terms:

999973?900002700912673

This confirms that the result is indeed 912673.

Using Long Multiplication (Non-Technical)

If we prefer not to use the binomial cube formula, we can still calculate 973 through long multiplication. Let's break it down step by step:

972100?39409 940997912673

Alternatively, we can directly multiply out 97×97×97 longhand:

" "97 × 97 9409
" "9409 × 97 912673

Lastly, we compare 912673 to 1000000, recognizing that 1003 is 106 (1,000,000), which is just a little more than 912673.