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Simplify the given equation ${\left( {256} \right)^{{{\left( 4 \right)}^{\dfrac{3}{2}}}}}$.

Last updated date: 13th Jun 2024
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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.