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Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.

#### a)

\n

Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.

\n

Using this rule,

\n

\\\begin{align} a^\\var{x} \\times a^\\var{y}\\ &= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\ &= a^\\var{x+y}. \\end{align} \

\n

#### b)

\n

We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.

\n

Notice there is a constant in front of each of the terms.

\n

To do this, write the product out explicitly, as

\n

\$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\$

\n

We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.

\n

Therefore:

\n

\\begin{align}
\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&= \\var{c*d} \\times a^\\var{p+q} \\\\
&= \\simplify{{c*d}*a^{p+q}}.
\\end{align}

\n

#### c)

\n

Here we are using: $a^m \\div a^n = a^{m-n}$.

\n

We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.

\n

To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as

\n

\$\\simplify{{b}*a^{x}/({g}*a^{y})} = \\simplify{{b}/{g}} \\times \\simplify{a^{x}/(a^{y})}.\$

\n

Using rules of indices,

\n

\\begin{align}                                                                                                                                                                                                                                                                                           \\frac{a^\\var{x}}{a^\\var{y}} &= a^\\var{x} \\div a^\\var{y}\\\\
&= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\
&= a^\\var{x-y}.
\\end{align}

\n

Therefore,

\n

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

\n

Alternatively,

\n

Using the rule of indices: $a^{-m} = \\displaystyle\\frac{1}{a^{m}}$, we can rewrite the question as:

\n

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\frac{a^\\var{x}}{a^\\var{y}}\\\\
&= \\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}}.
\\end{align}

\n

And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:

\n

\\begin{align}
\\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}} &= \\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

\n

#### d)

\n

The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.

\n

To do this we use the rules:

\n

\$(a^{m})^{n} = a^{mn},\$

\n

\$(ab)^m = a^mb^m.\$

\n

We can then expand the equation as

\n

\$(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\$

\n

Then using the rule of indices mentioned previously,

\n

\\\begin{align} (\\simplify{{c}*a^{p}})^{\\var{q}}&= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\ &= \\simplify{{c}^{q}*a^{p*q}}. \\end{align} \

\n

#### e)

\n

The question asks us to simplify $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.

\n

To do this we use the rules:

\n

\$a^\\frac{1}{m} = \\sqrt[m]{a},\$

\n

\$(ab)^m = a^mb^m.\$

\n

We can expand the expression as follows:

\n

\\\begin{align} \\sqrt[\\var{d}]{a} &= (\\simplify{a})^\\frac{1}{\\var{d}}\\\\ &= a^\\frac{1}{\\var{d}}. \\end{align} \

\n

#### f)

\n

The question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.

\n

Here, we use the rule of indices: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$, allowing us to expand the expression as follows:

\n

\\\begin{align} \\sqrt[\\var{c}]{\\simplify{a^{q}}} &= \\simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\\\ &= \\simplify[fractionnumbers,all]{a^{{q}/{c}}}. \\end{align} \

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Write $a^{\\var{x}} \\times a^{\\var{y}}$ as a single power of $a$.

\n

\n

$a^{\\var{x}} \\times a^{\\var{y}} =$ [[0]].

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Use the rule: $a^m \\times a^n = a^{m+n}$.

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Write $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$ as an integer multiplied by a single power of $a$.

\n

$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$ [[0]].

\n

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Write $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$ as a number multiplied by a single power of $a$.

\n

$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$ [[0]].

\n

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You could use one of the following rules:

\n

$a^m \\div a^n = a^{m-n}$.

\n

$a^{-m} = \\displaystyle\\frac{1}{a^m}$.

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Write $(\\simplify{{c}*a^{p}})^{\\var{q}}$ as an integer multiplied by a single power of $a$.

\n

$(\\simplify{{c}*a^{p}})^{\\var{q}} =$ [[0]].

\n

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Use the rules:

\n

$(ab)^m = a^mb^m$.

\n

$(a^m)^n = a^{mn}$.

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\n

Write $\\sqrt[\\var{d}]{a}$ as a single power of $a$.

\n

$\\sqrt[\\var{d}]{a} =$ [[0]].

\n

\n

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Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.

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Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.

\n

$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].

\n

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Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.

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This question aims to test understanding and ability to use the laws of indices.

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