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For an insight into an index of 0, consider the following table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

index form	$\\var{base}^3$	$\\var{base}^2$	$\\var{base}^1$	$\\var{base}^0$
result	$\\var{base^3}$	$\\var{base^2}$	$\\var{base}$	[[0]]

Notice each time the power decreases by $1$, the result is divided by $\\var{base}$. Using this idea, fill in the rest of the table.

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Each time you reduce the power by 1 you divide the result by the base, that is $\\var{base}$. Following this pattern:

\\[\\var{base}^{0}=\\frac{\\var{base}}{\\var{base}}=1\\]

From this we conclude that any non-zero number to the power of 0 is 1.

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$\\var{int1}^0$ = [[0]]

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Any non-zero number to the power of 0 is 1.

$\\var{dec1}^0$ = [[0]]

Any non-zero number to the power of 0 is 1.

$(\\var{nint1})^0$ = [[0]]

Any non-zero number to the power of 0 is 1.

$-(\\var{int2})^0$ = [[0]]

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Here $\\var{int2}$ is to the power of zero and so it evaluates to $1$. However, there is still a negative sign in front, so the answer must be $-1$.

Note the power of zero is only acting on what is inside the brackets.

$\\displaystyle\\left(\\simplify{{num1}/{den1}}\\right)^0$ = [[0]]

Any non-zero number to the power of 0 is 1.

$\\displaystyle\\frac{\\var{num2}^0}{\\var{den2}}$ = [[0]]

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Note that the power of zero is only acting on the numerator. As such the numerator is equal to $1$ and the denominator remains untouched.

$(\\var{int1}\\times\\var{int2})^0$ = [[0]]

Regardless of the result of the multiplication the power of zero will force it to evaluate to 1.

$\\var{smallint1}\\times\\var{smallint2}^0$ = [[0]]

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The power of zero is only acting on the second number in the product. It is only this number that will become 1.

Evaluate the following, without the use of a calculator:

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