Question

How do you simplify and write ${{\left( 4{{x}^{-4}}{{y}^{2}} \right)}^{-3}}$ with positive exponents?

Answer 1

Hint: In this question, we have to simplify an algebraic expression. Thus, we will use the basic mathematical rules and the exponent formula to get the required solution. First, we will open the bracket of the given problem by using the exponent rule ${{\left( ab \right)}^{x}}={{a}^{x}}{{b}^{x}}$ in the given algebraic expression. After that, we will again apply the exponent rule that says when we have a power of a power rule, then the powers are multiplied to each other, that is ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}$ . Then, we will again apply the formula ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$ and make the necessary calculations, to get the required solution for the problem.

Complete step by step solution:
According to the question, we have to simplify the given algebraic expression.
Thus, we will apply the basic mathematical rules and the exponent rule to get the solution.
The algebraic expression given to us is ${{\left( 4{{x}^{-4}}{{y}^{2}} \right)}^{-3}}$ ----------- (1)
So, we start solving our problem by applying the exponent rule ${{\left( ab \right)}^{x}}={{a}^{x}}{{b}^{x}}$ in equation (1), we get
$\left( {{4}^{-3}} \right){{\left( {{x}^{-4}} \right)}^{-3}}{{\left( {{y}^{2}} \right)}^{-3}}$
Now, we know the exponent rule, that states when we have a power of a power rule, then the powers are multiplied to each other, that is ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}$ . Thus, we will apply this formula in the above algebraic expression, we get
$\left( {{4}^{-3}} \right){{\left( x \right)}^{-4\times (-3)}}{{\left( y \right)}^{2\times (-3)}}$
On further solving the above algebraic expression, we get
$\left( {{4}^{-3}} \right){{\left( x \right)}^{12}}{{\left( y \right)}^{-6}}$
Now, we will again apply the algebraic formula ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$ in the above expression, we get
${{\left( \dfrac{1}{4} \right)}^{3}}{{\left( x \right)}^{12}}{{\left( \dfrac{1}{y} \right)}^{6}}$
On further simplification, we get
$\left( \dfrac{1}{64} \right){{\left( x \right)}^{12}}{{\left( \dfrac{1}{y} \right)}^{6}}$
Therefore, we get
$\dfrac{{{x}^{12}}}{64{{y}^{6}}}$
Thus, we cannot simplify the above expression.

Therefore, for the algebraic equation ${{\left( 4{{x}^{-4}}{{y}^{2}} \right)}^{-3}}$ , its simplified value is $\dfrac{{{x}^{12}}}{64{{y}^{6}}}$ which is the required solution with positive exponent.

Note: While solving this question, do step-by-step calculations to avoid mathematical errors. Also, do not forget that the powers are multiplied to each other instead of they are adding to each other. Do mention the formulas and the rules properly to get an accurate answer.