# How do you simplify -1^15?

Feb 12, 2016

#### Explanation:

$- {1}^{-} 15$

is (-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1) = -1

When the exponent of a negative root is odd the negative will multiply to a negative solution.

Feb 12, 2016

$- 1$

#### Explanation:

Given:$\text{ } {\left(- 1\right)}^{15}$

$\textcolor{b l u e}{\text{Consider the index (powers)}}$

$\text{Known that: } {\left(- 1\right)}^{2} = + 1$

But $\to \frac{15}{2} = 7 + \text{ Remainder of 1}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Using this to solve your problem}}$

${\left(- 1\right)}^{14} \times \left(- 1\right)$

$\textcolor{b r o w n}{\text{'---------------Something to consider----------------------------}}$

$\textcolor{b r o w n}{\text{Think for a moment about }}$

$\textcolor{b r o w n}{\left(- 1\right) \times \left(- 1\right) \times \left(- 1\right) \times \left(- 1\right) \times \left(- 1\right) \times \left(- 1\right)}$

$\textcolor{b r o w n}{\text{This is the same as } {\left(- 1\right)}^{2} \times {\left(- 1\right)}^{2} \times {\left(- 1\right)}^{2}}$

$\textcolor{b r o w n}{\text{Which is the same as " } {\left({\left(- 1\right)}^{2}\right)}^{3}}$
$\textcolor{b r o w n}{\text{Observe that the outermost index of 3 is "1/2" the original count. Also notice that } 2 \times 3 = 6}$
$\textcolor{b r o w n}{\text{'---------------------------------------------------------------------------}}$

$\text{ " ((-1)^2)^7xx"Remainder"" "->" "(+1)^7xx"Remainder}$

So we have$\text{ } {\left(+ 1\right)}^{7} \times \left(- 1\right) = - 1$

$\textcolor{g r e e n}{\text{If an index is odd, then the final value is negative}}$
$\textcolor{g r e e n}{\text{If an index is even then the final value is positive}}$

Feb 12, 2016

$- 1$

#### Explanation:

As there are no parentheses around the $- 1$, you need to evaluate this expression in the following order:

1) first compute the power: ${1}^{15} = 1$

2) afterwards negate the result from above: $- \left(1\right) = - 1$.

Thus, you need to compute:

$- {1}^{15} = - \left({1}^{15}\right) = - \left(1\right) = - 1$