Question

How do you simplify ${\left( {3{x^2}} \right)^{ - 3}}?$

Answer 1

Hint: To simplify the given exponential expression, one should use law of indices for negative powers, then use distributive property of exponent over multiplication and then apply law of indices for brackets to get the final simplified form of the given expression.

Complete step-by-step solution:
In order to simplify the given expression ${\left( {3{x^2}} \right)^{ - 3}}$ we will first use the law of indices for negative powers through which we will convert the negative power in to a positive one. And this is because the given expression is an exponential expression and in the simplified form of an exponential expression there is no negative powers, that’s why we will convert it into positive power as follows
With the help of law of indices for negative power, we can write the given expression as
${\left( {3{x^2}} \right)^{ - 3}} = \dfrac{1}{{{{\left( {3{x^2}} \right)}^3}}}$
Now using the distributive property of exponent over multiplication to distribute the exponent in individual terms or multiplicands as follows
$
  \dfrac{1}{{{{\left( {3{x^2}} \right)}^3}}} = \dfrac{1}{{{3^3}{{\left( {{x^2}} \right)}^3}}} \\
   = \dfrac{1}{{81{{\left( {{x^2}} \right)}^3}}} \\
 $
Now we will use law of indices for brackets to simplify further, law of indices for brackets tells that if an exponential term is in a bracket and bracket also have power then it is simplified as the base to the power product of both exponents
$\dfrac{1}{{81{{\left( {{x^2}} \right)}^3}}} = \dfrac{1}{{81{x^{2 \times 3}}}} = \dfrac{1}{{81{x^6}}}$

Therefore $\dfrac{1}{{81{x^6}}}$ is the simplified form of the given expression.

Note: When using the distributive property make sure the terms under the bracket either in multiplication or in division operation with each other, this is not applied in subtraction and addition. In other words, the distributive property of exponent holds good only for multiplication and division algebraic operations.