Question

What is \[{256^{\dfrac{3}{4}}}\]?

Answer 1

Hint: The given number is an algebraic expression which can be solved by converting the expression in simpler form. It contains fractional exponents. A fractional exponent is a technique for expressing powers and roots together. Exponents are powers or indices.

Complete step by step solution:
When you have a fractional exponent, the numerator is the power and the denominator is the root. To make a problem easier to solve you can break up the exponents by rewriting them.
The general form of a fractional exponent is:
\[{b^{\dfrac{n}{m}}} = {(\sqrt[m]{b})^n} = \sqrt[m]{{{b^n}}}\]
Where
\[b\]- Base : This is the number whose root is being calculated.
\[n\]-Power: The power determines how many times the value is root is multiplied by itself to get the base.
\[{b^n}\]- Radicand: The expression under the \[\sqrt {\text{ }} \] sign
\[m\]-Order or Index : The index or order of the radical is the number indicating the root being taken
After understanding the form of expression, we can now proceed to solve the sum as per above formula as follows:
\[ \Rightarrow {256^{\dfrac{3}{4}}} = {(\sqrt[4]{{256}})^3}\]
Removing the bracket and multiplying with the power,
\[ \Rightarrow {256^{\dfrac{3}{4}}} = \sqrt[4]{{{{256}^3}}}\]
The expression can further be simplified as follows:
\[ \Rightarrow {256^{\dfrac{3}{4}}} = \sqrt[4]{{{{(4 \times 4 \times 4 \times 4)}^3}}}\]
Solving the bracket and under root,
\[ \Rightarrow {256^{\dfrac{3}{4}}} = {4^3}\]
Solving to the power of we get,
\[ \Rightarrow {256^{\dfrac{3}{4}}} = 64\]
Therefore, the value of $256^{\dfrac{3}{4}}$ is \[64\].

Note:
1) We can solve this with alternate method as follows:
\[256\] can be written as \[{4^4}\]
Hence, we can write \[{256^{\dfrac{3}{4}}}\] as follows:
\[ \Rightarrow {256^{\dfrac{3}{4}}} = {({4^4})^{\dfrac{3}{4}}}\]
Multiplying the power of expression as per the rule \[{({a^m})^n} = {a^{mn}}\]
\[ \Rightarrow {256^{\dfrac{3}{4}}} = {4^{4 \times }}^{\dfrac{3}{4}}\]
Cancelling the common factor,
$\Rightarrow 256^{\dfrac{3}{4}}=4^{\not{4} \times \dfrac{3}{\not{4}}}$
We arrive at
\[ \Rightarrow {256^{\dfrac{3}{4}}} = {4^3}\]
\[ \Rightarrow {256^{\dfrac{3}{4}}} = 64\]
2) Multiplying terms having the same base and with fractional exponents is equal to adding together the exponents. For example, in the given sum, \[{4^1} \times {4^1} \times {4^1} \times {4^1} = {4^{1 + 1 + 1 + 1}} = {4^4}\].