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

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2..6

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$ w ^\\var{powers1[0][0]}\\times w ^\\var{powers1[1][0]} \\times w ^\\var{powers1[2][0]}$ = [[0]]

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Recall:

$w^\\var{powers1[0][0]}$ means $\\var{powers1[0][1]}$ $w$s all multiplied together.
$w^\\var{powers1[1][0]}$ means $\\var{powers1[1][1]}$ $w$s all multiplied together.
$w^\\var{powers1[2][0]}$ means $\\var{powers1[2][1]}$ $w$s all multiplied together.

So in total how many $w$s are there multiplied together?

Well, $\\var{powers1[0][0]}+\\var{powers1[1][0]}+\\var{powers1[2][0]}=\\var{sumpow1}$. And so our answer is $w^\\var{sumpow1}$.

Note, in general $a^ba^c=a^{b+c}$.

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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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$x^\\var{powers2[0][0]}\\times x^\\var{powers2[1][0]} \\times x^\\var{powers2[2][0]}$ = [[0]]

Recall:

$x^0$ means no $x$s all multiplied together. (As you know, $x^0=1$).
$x^1$ means just one $x$. (As you know, $x^1=x$)
$x^\\var{wild[0]}$ means $\\var{wild[1]}$ $x$s all multiplied together.

So in total how many $x$s are there multiplied together?

Well, $0+1+\\var{wild[0]}=\\var{sumpow2}$. And so our answer is $x^\\var{sumpow2}$.

Note, in general $a^ba^c=a^{b+c}$, $a^0=1$ and $a^1=a$.

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

$\\displaystyle a^\\var{dec}\\times a^\\var{neg} \\times a^{\\var{num}/\\var{den}}$ = [[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 ($a$) and we are multiplying, we can simply add the powers.

You could convert the fraction to a decimal and then add them all. Or you could add them all as fractions.

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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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$z^\\var{powers3[0][0]}\\times y^\\var{wild[0]} \\times z^\\var{powers3[1][0]}$ = [[0]]

Note: use * for multiplication.

It is important to note that the bases are different! We can only add the powers if the bases are the same.

Recall:

$z^\\var{powers3[0][0]}$ means $\\var{powers3[0][1]}$ $z$s all multiplied together.
$z^\\var{powers3[1][0]}$ means $\\var{powers3[1][1]}$ $z$s all multiplied together.
$y^\\var{wild[0]}$ means $\\var{wild[1]}$ $y$s all multiplied together.

So in total what do we have?

$y^\\var{wild[0]}z^\\var{sumpow4}$.

Note we would type y^{wild[0]}*z^{sumpow4}.

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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?

$m\\times p^\\var{powers1[0][0]} = (mp)^\\var{powers1[0][0]}$

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?

$q\\times u^\\var{powers1[0][0]} = (qu)^\\var{powers1[0][0]+1}$

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 the powers if the bases are the same.

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