
How do you simplify $\dfrac{{{144}^{14}}}{{{144}^{2}}}?$
Answer
445.5k+ views
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.
(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.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What constitutes the central nervous system How are class 10 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE
