Answer

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**Hint:**In the given example, you can find the value of the given fraction by simplifying the fraction using property of laws of indices you will require following properties for solving this problem.

(1) $\dfrac{1}{{{a}^{m}}}={{a}^{-m}}$

(2) ${{a}^{m}}\times {{a}^{n}}={{a}^{m-n}}$

(3) ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$

**Complete step by step solution:**

In the given example we have to find the value of given fraction to simplify the given fraction that is $\dfrac{{{144}^{14}}}{{{144}^{2}}}$

We can write this as,

${{144}^{14}}\times \dfrac{1}{{{144}^{2}}}$

Let us simplify firstly the term $\dfrac{1}{{{144}^{2}}}$

Another way of writing the above term is ${{144}^{-2}}$ as we can write this by using the law of indices.

Now, the expression can be written as,

${{144}^{14}}\times {{144}^{-2}}$

As both numbers are $144$ i.e. the base value is the same.

Therefore you can write this by the property of ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$

As,

${{144}^{\left( 14-2 \right)}}$

Now, after subtracting $2$ from $14$ we get $12.$ So, we have ${{144}^{12}}$

Also, we know that ${{12}^{2}}=144$

Therefore, above value can be modified as,

${{\left( {{12}^{2}} \right)}^{12}}$

Now, multiplying the power $2$ and $12$ as by ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ this property we can multiply $2$ and $12$

So, we get,

${{12}^{24}}$

**Hence, the required solution of $\dfrac{{{144}^{14}}}{{{144}^{2}}}$ is ${{12}^{24}}$**

**Additional Information:**

This is an example of powers of fractions. It involves multiplying the number by itself the number of times indicated by the exponent.

A fractional value represents a part of a whole or more generally any number of equal parts. Here ${{12}^{24}}$ is represents in the form of fraction that is $\dfrac{{{144}^{14}}}{{{144}^{2}}}$

Following are some properties of laws of indices.

(1) ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$

(2) ${{a}^{-m}}={{a}^{m-1}}=\dfrac{1}{{{a}^{m-1}}}$

(3) ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$

(4) $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}=\dfrac{1}{{{a}^{m-1}}}$

(5) ${{(ab)}^{n}}={{a}^{n}}.{{b}^{n}}$

(6) ${{a}^{0}}=1$

(7) If ${{a}^{m}}={{a}^{n}}$ then $m=n$

(8) If ${{a}^{n}}={{b}^{n}},a\ne b$ then $n=0$

**Note:**Apply the properties of law of indices carefully and simplify it. For this you have to know which property can be applied on which step. Using the property of law of indices you will easily get the required solution. Also, solving this problem using property will save your time as well.

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