# How do you evaluate 4.246\times 4.246?

Dec 15, 2016

$18.028516$

#### Explanation:

First we rewrite these numbers as fractions and simplify there denominators if possible.

$4.246 = \frac{4246}{1000} = \frac{4246}{10} ^ 3$

So $4.246 \cdot 4.246 = \frac{4246}{10} ^ 3 \cdot \frac{4246}{10} ^ 3$ can be rewritten as $\frac{4246 \cdot 4264}{{10}^{3} \cdot {10}^{3}}$.

Two equal numbers set to a power can be multiplied by setting that same number to the power of the two original powers added together. ${10}^{3} \cdot {10}^{3} = {10}^{3 + 3} = {10}^{6}$.

So now we are left with: $\frac{4246 \cdot 4246}{10} ^ 6$.

$4246 \cdot 4246$ can be rewritten as $\left(4000 + 200 + 40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)$.

Now we can solve using simple multiplication and addition:$\frac{\left(4000 + 200 + 40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$.

$\frac{\left(4000 + 200 + 40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{\left(4000 \cdot 4000\right) + \left(4000 \cdot 200\right) + \left(4000 \cdot 40\right) + \left(4000 \cdot 6\right) + \left(200 + 40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$.

$= \frac{16984000 + \left(200 + 40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{16984000 + \left(200 \cdot 4000\right) + \left(200 \cdot 200\right) + \left(200 \cdot 40\right) + \left(200 \cdot 6\right) + \left(40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{16984000 + 849200 + \left(40 + 6\right) \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{16984000 + 849200 + \left(40 \cdot 4000\right) + \left(40 \cdot 200\right) + \left(40 \cdot 40\right) + \left(40 \cdot 6\right) + 6 \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{16984000 + 849200 + 169840 + 6 \cdot \left(4000 + 200 + 40 + 6\right)}{10} ^ 6$

$= \frac{16984000 + 849200 + 169840 + \left(6 \cdot 4000\right) + \left(6 \cdot 200\right) + \left(6 \cdot 40\right) + \left(6 \cdot 6\right)}{10} ^ 6$

$= \frac{16984000 + 849200 + 169840 + 25476}{10} ^ 6$

$= \frac{18028516}{10} ^ 6$

Now the last step is to divided by 10^6 which is equal to adding 6 decimal places to a number.

$= \frac{18028516}{10} ^ 6 = 18.028516$