Answer
405k+ views
Hint: Use the various exponent rules to simplify the given expression. Start with using ${a^{\dfrac{n}{m}}} = \sqrt[m]{{{a^n}}}$ to simplify the power of the expression, i.e. ${4^{\dfrac{3}{2}}}$ . After simplifying the power, change the base $256$ in exponential form by finding the prime factors.
Complete step-by-step answer:
Here in the problem, we are given an expression ${\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}}$ , involving exponents and powers. And using properties and identities in exponents, we need to simplify this given expression.
Exponents and powers are ways used to represent very large numbers or very small numbers in a simplified manner. Power is an expression that shows repeated multiplication of the same number or factor. The value of the exponent is based on the number of times the base is multiplied to itself.
We often call exponents powers or indices. In other words, power refers to an expression that represents repeated multiplication of the same number while exponent is a quantity that represents the power to which we raise the number. Basically, we often use both these terms interchangeably in mathematical operations.
Before starting with the solution we should understand a few identities in exponents:
According to the definition of power, we can say: ${a^n} = a \times a \times a \times a.......n{\text{ }}times$ ……(i)
Also, ${a^{\dfrac{1}{m}}} = \sqrt[m]{a}{\text{ Therefore, }} \Rightarrow {a^{\dfrac{n}{m}}} = \sqrt[m]{{{a^n}}}$ …..(ii)
${a^{{{\left( m \right)}^n}}} = {a^{m \times m \times m.....n{\text{ }}times}}$ ……(iii)
The given expression is of the form ${\left( a \right)^{{{\left( b \right)}^{\dfrac{m}{n}}}}}$ and can be easily simplified using (ii) on the power, we get:
$ \Rightarrow {\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {\left( {256} \right)^{\sqrt {{4^3}} }}$
Now the number inside the radical sign can be simplified using relationship (i), this will give us:
$ \Rightarrow {\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {\left( {256} \right)^{\sqrt {{4^3}} }} = {\left( {256} \right)^{\sqrt {4 \times 4 \times 4} }}$
Since all the numbers inside the radical sign are positive, we can use $\sqrt {m \times n \times p} = \sqrt m \times \sqrt n \times \sqrt p $
\[ \Rightarrow {\left( {256} \right)^{\sqrt {4 \times 4 \times 4} }} = {\left( {256} \right)^{\sqrt 4 \times \sqrt 4 \times \sqrt 4 }}\]
Cleary the square root of a number $4$ is equal to $2$ , on substituting this value, we get:
\[ \Rightarrow {\left( {256} \right)^{\sqrt 4 \times \sqrt 4 \times \sqrt 4 }} = {\left( {256} \right)^{2 \times 2 \times 2}} = {\left( {256} \right)^8}\]
We can express the base in form of prime factors, i.e. $256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^8}$
Therefore, we get:
\[ \Rightarrow {\left( {256} \right)^8} = {\left( {{2^8}} \right)^8}\]
This can be further simplified using the relation ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$
Thus, we get the required expression as:
\[ \Rightarrow {\left( {256} \right)^8} = {\left( {{2^8}} \right)^8} = {2^{8 \times 8}} = {2^{64}}\]
Thus, we simplified the given expression as: ${\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {2^{64}}$.
Note: In questions like this correct use of exponents rule always plays a crucial role in the solution of the problem. Notice that the expression ${\left( a \right)^{{{(b)}^m}}}$ and expression ${\left( {{a^b}} \right)^m}$ are different since ${\left( a \right)^{{{(b)}^m}}} = {a^{b \times b \times b......m{\text{ }}times}}$ but for the other ${\left( {{a^b}} \right)^m} = {a^{b \times m}}$ . Be careful while simplifying such expressions with parentheses.
Complete step-by-step answer:
Here in the problem, we are given an expression ${\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}}$ , involving exponents and powers. And using properties and identities in exponents, we need to simplify this given expression.
Exponents and powers are ways used to represent very large numbers or very small numbers in a simplified manner. Power is an expression that shows repeated multiplication of the same number or factor. The value of the exponent is based on the number of times the base is multiplied to itself.
We often call exponents powers or indices. In other words, power refers to an expression that represents repeated multiplication of the same number while exponent is a quantity that represents the power to which we raise the number. Basically, we often use both these terms interchangeably in mathematical operations.
Before starting with the solution we should understand a few identities in exponents:
According to the definition of power, we can say: ${a^n} = a \times a \times a \times a.......n{\text{ }}times$ ……(i)
Also, ${a^{\dfrac{1}{m}}} = \sqrt[m]{a}{\text{ Therefore, }} \Rightarrow {a^{\dfrac{n}{m}}} = \sqrt[m]{{{a^n}}}$ …..(ii)
${a^{{{\left( m \right)}^n}}} = {a^{m \times m \times m.....n{\text{ }}times}}$ ……(iii)
The given expression is of the form ${\left( a \right)^{{{\left( b \right)}^{\dfrac{m}{n}}}}}$ and can be easily simplified using (ii) on the power, we get:
$ \Rightarrow {\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {\left( {256} \right)^{\sqrt {{4^3}} }}$
Now the number inside the radical sign can be simplified using relationship (i), this will give us:
$ \Rightarrow {\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {\left( {256} \right)^{\sqrt {{4^3}} }} = {\left( {256} \right)^{\sqrt {4 \times 4 \times 4} }}$
Since all the numbers inside the radical sign are positive, we can use $\sqrt {m \times n \times p} = \sqrt m \times \sqrt n \times \sqrt p $
\[ \Rightarrow {\left( {256} \right)^{\sqrt {4 \times 4 \times 4} }} = {\left( {256} \right)^{\sqrt 4 \times \sqrt 4 \times \sqrt 4 }}\]
Cleary the square root of a number $4$ is equal to $2$ , on substituting this value, we get:
\[ \Rightarrow {\left( {256} \right)^{\sqrt 4 \times \sqrt 4 \times \sqrt 4 }} = {\left( {256} \right)^{2 \times 2 \times 2}} = {\left( {256} \right)^8}\]
We can express the base in form of prime factors, i.e. $256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^8}$
Therefore, we get:
\[ \Rightarrow {\left( {256} \right)^8} = {\left( {{2^8}} \right)^8}\]
This can be further simplified using the relation ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$
Thus, we get the required expression as:
\[ \Rightarrow {\left( {256} \right)^8} = {\left( {{2^8}} \right)^8} = {2^{8 \times 8}} = {2^{64}}\]
Thus, we simplified the given expression as: ${\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}} = {2^{64}}$.
Note: In questions like this correct use of exponents rule always plays a crucial role in the solution of the problem. Notice that the expression ${\left( a \right)^{{{(b)}^m}}}$ and expression ${\left( {{a^b}} \right)^m}$ are different since ${\left( a \right)^{{{(b)}^m}}} = {a^{b \times b \times b......m{\text{ }}times}}$ but for the other ${\left( {{a^b}} \right)^m} = {a^{b \times m}}$ . Be careful while simplifying such expressions with parentheses.
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)