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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}^{-1}=\\frac{1}{\\var{base}}\\]

and

\\[\\var{base}^{-2}=\\frac{1}{\\var{base}}\\div\\var{base}=\\frac{1}{\\var{base}^2}\\]

Note you could input the second expression as $1/\\var{base}/\\var{base}$ or $1/\\var{base}\\wedge2$ or $1/\\var{base^2}$.

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Don't use negative powers here in this table.

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Don't use negative powers here in this table.

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For an insight into negative indices, consider the following table:

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

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

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

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Notice:
\\[1=\\var{base1}^0=\\var{base1}^{\\var{power1}-\\var{power1}}=\\var{base1}^{\\var{power1}} \\var{base1}^{-\\var{power1}}\\]

That is, we have

\\[1=\\var{base1}^{\\var{power1}} \\var{base1}^{-\\var{power1}}\\]

by dividing both sides of this equation by $\\var{base1}^{\\var{power1}}$ we get

\\[\\frac{1}{\\var{base1}^{\\var{power1}}}=\\var{base1}^{-\\var{power1}}\\]

In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$.

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Use ^ for powers. Input your answer in index form.

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Write your answer with a negative index.

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The fraction $\\displaystyle\\frac{1}{\\var{base1}^\\var{power1}}$ can be written using a negative index as [[0]].

Notice:
\\[1=\\var{base2}^0=\\var{base2}^{\\var{power2}+\\var{-power2}}=\\var{base2}^{\\var{power2}} \\var{base2}^{\\var{-power2}}\\]

That is, we have

\\[1=\\var{base2}^{\\var{power2}} \\var{base2}^{\\var{-power2}}\\]

by dividing both sides of this equation by $\\var{base2}^{\\var{-power2}}$ we get

\\[\\frac{1}{\\var{base2}^{\\var{-power2}}}=\\var{base2}^{\\var{power2}}\\]

In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$.

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Use ^ for powers. Input your answer in index form.

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The expression $\\var{base2}^{\\var{power2}}$ can be written without a negative index as the fraction [[0]].

You can think of the negative power as forcing the term to the other part of the fraction (if it was on top it goes to the bottom and if it was on the bottom it goes to the top) except now it has a positive power.

We can see this by using our rules for dividing fractions:

\\[\\frac{1}{\\var{base3}^\\var{-power3}}=\\frac{1}{\\left(\\frac{1}{\\var{base3}^\\var{power3}}\\right)}=1\\div{\\left(\\frac{1}{\\var{base3}^\\var{power3}}\\right)}=1\\times \\left(\\frac{{\\var{base3}^\\var{power3}}}{1}\\right)=\\var{base3}^\\var{power3}\\]

In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$ and $\\displaystyle\\frac{1}{a^{-c}}=a^{c}$.

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Use index form.

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Don't use fractions or negative indices.

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The expression $\\displaystyle\\frac{1}{\\var{base3}^{\\var{-power3}}}$ can be written without a negative index and without the use of a fraction.

The simplest way to write $\\displaystyle\\frac{1}{\\var{base3}^{\\var{-power3}}}$ in index form would be [[0]]

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Use ^ to signify powers.

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2..6

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