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Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers. Use negative powers if necessary.

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2..6

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$\\var{base1}^\\var{powers1[0][0]}\\div \\var{base1}^\\var{powers1[1][0]}$ = [[0]]

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Write the division as a fraction and cancel common factors or use the index law $\\displaystyle\\frac{a^b}{a^c}=a^{b-c}$.

Note, negative indices mean you divided by more than you had, for example, $\\displaystyle \\frac{1}{12^3}$ can be written as $12^{-3}$.

For our particular question we have, $\\var{base1}^\\var{powers1[0][0]}\\div \\var{base1}^\\var{powers1[1][0]}=\\frac{\\var{base1}^\\var{powers1[0][0]}}{\\var{base1}^\\var{powers1[1][0]}}$ is equal to $\\var{base1}^{\\var{powers1[0][0]}-\\var{powers1[1][0]}}=\\var{base1}^{\\var{diffpow1}}$.

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

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

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$\\displaystyle\\frac{\\var{base2}^\\var{powers2[0]}\\times\\var{base2}}{\\var{base2}^\\var{powers2[1]} \\times \\var{base2}^\\var{powers2[2]}}$ = [[0]]

Cancel common factors or use the index laws, e.g. $\\displaystyle\\frac{a^b}{a^c}=a^{b-c}$ and $a^b\\times a^c = a^{b+c}$.

Note, negative indices mean you divided by more than you had, for example, $\\displaystyle \\frac{1}{12^3}$ can be written as $12^{-3}$.

For $\\frac{\\var{base2}^\\var{powers2[0]}\\times\\var{base2}}{\\var{base2}^\\var{powers2[1]} \\times \\var{base2}^\\var{powers2[2]}}$ we will simplify the numerator and denominator by using $a^b\\times a^c = a^{b+c}$ and end up with $\\frac{\\var{base2}^\\var{powers2[0]+1}}{\\var{base2}^\\var{powers2[1]+powers2[2]}}$. Then we will use $\\frac{a^b}{a^c}=a^{b-c}$ to finally get $\\var{base2}^\\var{powers2[0]+1-powers2[1]-powers2[2]}$.

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

Use the same approach you used in the above questions to simplify the following in index form.

$\\displaystyle\\frac{\\var{base2}^\\var{ndec}}{\\var{base2}^\\var{neg}}$ = [[0]]

Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.

Since the bases are all the same ($\\var{base2}$) and we are dividing, we can subtract the powers since in general, we have $\\displaystyle\\frac{a^b}{a^c}=a^{b-c}$. Also recall that subtracting a negative is equivalent to adding a positive.

$\\begin{align}\\frac{\\var{base2}^\\var{ndec}}{\\var{base2}^\\var{neg}}&=\\var{base2}^{\\var{ndec}-\\var{neg}}\\\\
&=\\var{base2}^{\\var{ndec}+\\var{abs(neg)}}\\\\&=\\var{base2}^{\\var{ndec+abs(neg)}}\\end{align}$

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

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

Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers. Use negative powers if necessary.

$\\var{base3}^\\var{powers3[0][0]}\\times \\var{base1}^\\var{powers3[1][0]} \\div \\var{base3}^\\var{powers3[2][0]}$ = [[0]]

Note: use * for multiplication.

It is important to note that the bases are different! Index laws can only be applied if the bases are the same (or can be made the same). Because of this, we deal with the different bases separately.

Notice the first part of the expression can not be simplified using index laws.

$\\var{base3}^\\var{powers3[0][0]}\\times \\var{base1}^\\var{powers3[1][0]}$

However, with the division we can do some simplification. We can either:

write it as a fraction and cancel the common factor of $\\var{base3}^\\var{minpow4}$ from the top and bottom:
\\[\\frac{\\var{base3}^\\var{powers3[0][0]}\\times \\var{base1}^\\var{powers3[1][0]}}{ \\var{base3}^\\var{powers3[2][0]}}=\\frac{\\var{base3}^\\var{powers3[0][0]-minpow4}\\times \\var{base1}^\\var{powers3[1][0]}}{ \\var{base3}^\\var{powers3[2][0]-minpow4}}=\\var{base3}^\\var{diffpow4}\\var{base1}^\\var{powers3[1][0]}\\]
Subtract the powers, \"top power minus the bottom power\" for the terms with the same base:
\\[\\frac{\\var{base3}^\\var{powers3[0][0]}\\times \\var{base1}^\\var{powers3[1][0]}}{ \\var{base3}^\\var{powers3[2][0]}}=\\var{base3}^\\var{diffpow4}\\var{base1}^\\var{powers3[1][0]}\\]

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

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

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Is the following statement true or false?

$\\displaystyle\\frac{\\var{2*base2}}{\\var{base2}^\\var{powers1[0][0]}} = \\var{2}^\\var{-powers1[0][0]}$

It is important to note that the bases are different! Index laws only can be applied if the bases are the same (or can be made the same).

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Is the following statement true or false?

$\\displaystyle\\frac{\\var{2*base2}}{\\var{base2}^\\var{powers1[0][0]}} = \\var{2}^\\var{1-powers1[0][0]}$

It is important to note that the bases are different! Index laws only can be applied if the bases are the same (or can be made the same). We can only add (or subtract) the powers if the bases are the same.

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Is the following statement true or false?

$\\displaystyle\\frac{\\var{2*base2}}{\\var{base2}^\\var{powers1[0][0]}} = 2\\times\\var{base2}^\\var{1-powers1[0][0]}$

It is important to note that the bases are different! Index laws can only be applied if the bases are the same (or can be made the same). We can only add (or subtract) the powers if the bases are the same.

Note in this question we can make the bases the same.

\\[\\frac{\\var{2*base2}}{\\var{base2}^\\var{powers1[0][0]}} = \\frac{2\\times\\var{base2}}{\\var{base2}^\\var{powers1[0][0]}} = 2\\times\\var{base2}^\\var{1-powers1[0][0]}\\]

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