Question

How do you simplify ${{\left( {{n}^{4}} \right)}^{\dfrac{3}{2}}}$

Answer 1

Hint: Now consider the given expression ${{\left( {{n}^{4}} \right)}^{\dfrac{3}{2}}}$ . Now we know by exponent rule of indices that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ hence using this we will first simplify the expression. Then on simplifying the terms in power we will get the required value of the given expression.

Complete step by step solution:
To solve the given problem we must first understand the concept of indices.
Now indices are nothing but a number or a variable raised to some power.
Let us understand the concept by an example.
Let us say we have a number ${{3}^{2}}$ now here we have a number 3 and the power raised to 3 is 2.
Now the power represents how n=many times a number is multiplied by itself. Now since in the example the power is given as 2, 3 must be multiplied by itself 2 times.
Hence we have ${{3}^{2}}=3\times 3$ .
Now with the indices we have some basic laws of indices.
Let us first understand the law of multiplication.
According to this we have ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ .
Now next let us understand the law of division. This law states $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Now next let us understand the law of exponent ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
These laws can be easily verified by taking different values of m and n for any a.
Now let us consider the given expression ${{\left( {{n}^{4}} \right)}^{\dfrac{3}{2}}}$.
Now using the law of exponent we get ${{n}^{4\times \dfrac{3}{2}}}$ .
Now simplifying the above expression we get,
$\Rightarrow {{n}^{3\times 2}}={{n}^{6}}$

Hence the value of the given expression is ${{n}^{6}}$ .

Note: Now note that the power of a number or a variable can be any real number. Hence it can be positive, negative, fraction or 0 also. Also note that for any a which is real we have ${{a}^{0}}=1$ .
Similarly we also have ${{a}^{1}}=a$ hence if we have nothing written in power we can take it as 1.