Answer
414.9k+ views
Hint: Here, use the basic formula of the difference of two squares and the property of the splitting of the powers using the multiplicative identity. Also, use the fundamental of the squares and square roots which cancels each other.
Complete step-by-step answer:
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}\] ………….. (a)
Simplify using the formula the difference of two squares –
${{a}^{2}}-{{b}^{2}}=(a-b)(a+b)$
\[\begin{align}
\Rightarrow & {{\left( {{61}^{2}}-{{11}^{2}} \right)}^{{}}}=(61-11)(61+11) \\
\Rightarrow & \left( {{61}^{2}}-{{11}^{2}} \right)=(50)(72) \\
\Rightarrow & \left( {{61}^{2}}-{{11}^{2}} \right)=3600\,\text{ }..........\text{(b)} \\
\end{align}\](Simplify)
Place the value of equation (b) in the equation (a) –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{(3600)}^{\dfrac{3}{2}}}\] …………..(c)
Simplify, using the basic mathematical operations –
Split the power –
$\Rightarrow$ $\dfrac{3}{2}=\left( \dfrac{1}{2} \right)\times 3$
Place the above value in the equation (c) –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ {{(3600)}^{\dfrac{1}{2}}} \right]}^{3}}\]
Simplify the left hand side of the equation –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ {{({{60}^{2}})}^{\dfrac{1}{2}}} \right]}^{3}}\]
By the property – the squares and square-root cancels each other –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ (60) \right]}^{3}}\]
Now, simplify the left hand side of the equation and do the cubes of six and ten -
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}=216000\]
Therefore, the required solution is \[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}=216000\]
Note: The square root of the number “n” is the number when multiplied by the number itself and equals to “n”. For example, the square root of $\sqrt{9}=\sqrt{{{3}^{2}}}=3$. The squares and the square roots are opposite to each other and so cancel each other. Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $\text{16 = 4 }\times \text{ 4, 16=}{{\text{4}}^{2}}$, generally it is denoted by n to the power two i.e. ${{n}^{2}}$. The perfect square is the number which can be expressed as the product of the two equal integers. For example: $9$, it can be expressed as the product of equal integers. $9=3\times 3$. Cube is the number, we get when the number is multiplied three times. For example - ${{n}^{3}}=n\times n\times n$
${{3}^{3}}=3\times 3\times 3,\ \text{implies }{{3}^{3}}=27$
Complete step-by-step answer:
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}\] ………….. (a)
Simplify using the formula the difference of two squares –
${{a}^{2}}-{{b}^{2}}=(a-b)(a+b)$
\[\begin{align}
\Rightarrow & {{\left( {{61}^{2}}-{{11}^{2}} \right)}^{{}}}=(61-11)(61+11) \\
\Rightarrow & \left( {{61}^{2}}-{{11}^{2}} \right)=(50)(72) \\
\Rightarrow & \left( {{61}^{2}}-{{11}^{2}} \right)=3600\,\text{ }..........\text{(b)} \\
\end{align}\](Simplify)
Place the value of equation (b) in the equation (a) –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{(3600)}^{\dfrac{3}{2}}}\] …………..(c)
Simplify, using the basic mathematical operations –
Split the power –
$\Rightarrow$ $\dfrac{3}{2}=\left( \dfrac{1}{2} \right)\times 3$
Place the above value in the equation (c) –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ {{(3600)}^{\dfrac{1}{2}}} \right]}^{3}}\]
Simplify the left hand side of the equation –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ {{({{60}^{2}})}^{\dfrac{1}{2}}} \right]}^{3}}\]
By the property – the squares and square-root cancels each other –
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}={{\left[ (60) \right]}^{3}}\]
Now, simplify the left hand side of the equation and do the cubes of six and ten -
\[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}=216000\]
Therefore, the required solution is \[{{\left( {{61}^{2}}-{{11}^{2}} \right)}^{\dfrac{3}{2}}}=216000\]
Note: The square root of the number “n” is the number when multiplied by the number itself and equals to “n”. For example, the square root of $\sqrt{9}=\sqrt{{{3}^{2}}}=3$. The squares and the square roots are opposite to each other and so cancel each other. Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $\text{16 = 4 }\times \text{ 4, 16=}{{\text{4}}^{2}}$, generally it is denoted by n to the power two i.e. ${{n}^{2}}$. The perfect square is the number which can be expressed as the product of the two equal integers. For example: $9$, it can be expressed as the product of equal integers. $9=3\times 3$. Cube is the number, we get when the number is multiplied three times. For example - ${{n}^{3}}=n\times n\times n$
${{3}^{3}}=3\times 3\times 3,\ \text{implies }{{3}^{3}}=27$
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)