Question

How do you simplify ${{\left( 2{{b}^{4}} \right)}^{-1}}$ and write it using the positive exponents?

Answer 1

Hint: In this question, we have to simplify an algebraic function. So, we will use the basic mathematical rules and the exponent formula to find the solution to the problem. We will first apply the exponent formula \[{{\left( a\times b \right)}^{c}}={{a}^{c}}\times {{b}^{c}}\] in the given algebraic function. Then, we will again apply the exponent formula ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{b\times c}}$ in the new algebraic function. After simplification, we will again use the exponent formula ${{a}^{-1}}=\dfrac{1}{a}$ in the new algebraic function to get the required result for the problem.

Complete step-by-step answer:
According to the problem, we have to simplify an algebraic function.
Thus, we will use the exponent formula to get the result.
The algebraic function given to us is ${{\left( 2{{b}^{4}} \right)}^{-1}}$ ------- (1)
Therefore, we will first apply the formula \[{{\left( a\times b \right)}^{c}}={{a}^{c}}\times {{b}^{c}}\] in equation (1), we get
\[{{2}^{-1}}\times {{\left( {{b}^{4}} \right)}^{(-1)}}\]
Now, we will again apply the exponent formula ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{b\times c}}$ in the above equation, we get
\[{{2}^{-1}}\times {{\left( b \right)}^{4\times (-1)}}\]
On further simplification, we get
\[{{2}^{-1}}\times {{b}^{-4}}\]
Now, we will again use the exponent formula ${{a}^{-1}}=\dfrac{1}{a}$ in the above equation, we get
\[\dfrac{1}{2}\times \dfrac{1}{{{b}^{4}}}\]
Therefore, on more simplification, that is using the basic mathematical rule $\dfrac{1}{a}\times \dfrac{1}{b}=\dfrac{1}{ab}$ in the above function, we get
\[\dfrac{1}{2{{b}^{4}}}\] which is our required answer.
Therefore, for the algebraic function ${{\left( 2{{b}^{4}} \right)}^{-1}}$, its simplified value with a positive exponent is equal to \[\dfrac{1}{2{{b}^{4}}}\] .

Note: While solving this problem, keep in mind the formulas you are using. Always mention them in the steps wherever it is applicable to avoid confusion and mathematical errors. At the time of applying the first exponent formula, we get \[{{2}^{-1}}\times {{\left( {{b}^{4}} \right)}^{(-1)}}\] , so do not confuse it with the power 4, it is only applicable on variable b and not on number 2.