Answer

Verified

340.2k+ views

**Hint:**Here an expression is given which has a fraction exponent.

For solving fraction exponent, we have to use some general rules of fractional exponent.

As for ${{n}^{th}}$ root and $m$ power for number $a$, the general form can be written as,

\[{{a}^{\dfrac{m}{n}}}={{\left( \sqrt[n]{a} \right)}^{m}}\]

Where,

\[n=\] root

\[m=\] power

By using the above fraction rule we can solve the given expression.

**Complete step by step solution:**Given that, there is an expression having fractional exponent as,

${{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}$, and we have to solve it,

We know the same general rules for fraction exponent which is as follows,

\[{{x}^{-1}}=\dfrac{1}{x}\]

And \[{{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}}\]

Now using above general rules we have to solve the given expression

${{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}$ (given)

Now use the first rule which we have already written,

i.e. \[{{x}^{-1}}=\dfrac{1}{x}\]

\[\therefore {{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}={{\left( 16 \right)}^{\dfrac{3}{4}}}\]

But, we know that, \[{{\left( 2 \right)}^{4}}=16\]

So, now use the second rule which we have written

i.e. \[{{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}}\]

\[\therefore {{\left( {{2}^{4}} \right)}^{\dfrac{3}{4}}}={{2}^{{4}\times \dfrac{3}{{{4}}}}}\]

\[={{2}^{3}}\]

And the expression becomes

\[{{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}={{2}^{3}}\,\,=\,8\]

**Additional Information:**

Expressing the power and roots together is known as fractional exponent.

The general form of writing the fractional exponent for \[{{n}^{th}}\] root is as follows:

\[\therefore {{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\]

It means that, when \[{{n}^{th}}\] root of \[a\] is multiplied by \[n\] times, it will give the result as \[a\]

\[{{a}^{\dfrac{1}{n}}}\times {{a}^{\dfrac{1}{n}}}\times {{a}^{\dfrac{1}{n}}}\times .......\times {{a}^{\dfrac{1}{n}}}=a\]

For example \[{{625}^{\dfrac{1}{4}}}\]

\[\therefore {{625}^{\dfrac{1}{4}}}=\sqrt[4]{625}\]

As we know that, \[{{5}^{4}}=625\]

Therefore, the answer will be

\[{{625}^{\dfrac{1}{4}}}=\sqrt[4]{625}=\sqrt[4]{{{5}^{4}}}=5\]

In the above equation \[625\]is radicand as it is under the radical sign.

The order of the above equation is \[4\] as it indicates the root. Now, for \[{{n}^{th}}\] root and \[m\]power of the general form ca =n be \[{{a}^{3}}\]

\[{{a}^{\dfrac{m}{n}}}={{\left( \sqrt[n]{a} \right)}^{m}}\]

It means that we have to take the root of \[a\] as \[n\] and power as \[m\] as, \[{{\left( {{a}^{\dfrac{1}{n}}} \right)}^{m}}\]

**Note:**

In this numerical, the given expression has a fractional exponent. So, we have to solve it by using the rules of fraction exponent. The given expression can solve in other way also which is as follows

${{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}$ given

But as we know that,

\[{{x}^{-1}}=\dfrac{1}{x}\]

\[\therefore {{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}={{(16)}^{\dfrac{3}{4}}}\]

Now according to general form of \[{{n}^{th}}\] root and \[m\] power for number \[a\] is,

\[{{a}^{\dfrac{m}{n}}}={{\left( \sqrt[n]{a} \right)}^{m}}\]

So, from above general form compare to \[{{(16)}^{\dfrac{3}{4}}}\] we have \[4\] as a root and \[3\] as a power

i.e. \[n=4;\,m=3\]

Therefore, the equation becomes,

\[{{\left( 16 \right)}^{\dfrac{3}{4}}}={{\left( \sqrt[4]{16} \right)}^{3}}\]

As we know that, \[{{2}^{4}}=16\]

Therefore, \[{{\left( 16 \right)}^{\dfrac{3}{4}}}={{\left( \sqrt[4]{24} \right)}^{3}}\]

\[={{\left( 2 \right)}^{3}}=8\]

Therefore, the final solution for given expression \[{{\left( \dfrac{1}{16} \right)}^{\dfrac{-3}{4}}}\] is \[8\]

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Distinguish between the reserved forests and protected class 10 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give simple chemical tests to distinguish between the class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Which of the following books is not written by Harshavardhana class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

In which states of India are mango showers common What class 9 social science CBSE

What Made Mr Keesing Allow Anne to Talk in Class class 10 english CBSE