Question

How do you simplify $ {( - 2{x^2})^3}{(4{x^3})^{ - 1}}? $

Answer 1

Hint: We know that the above given question is in exponential form. An exponent refers to the number of times a number is multiplied by itself. There is base and exponent or power in this type of equation. Here, in the given question $ ( - 2{x^2}) $ and $ (4{x^3}) $ are the base and the number $ 3 $ and $ - 1 $ are the exponential power. As we know that as per the property of exponent rule if there is $ \dfrac{{{a^m}}}{{{a^n}}} $ then it can be written as $ {a^{m - n}} $ . When we express a number in exponential form then we can say that it’s power has been raised by the exponent.

Complete step-by-step solution:
There is one basic exponential rule that is commonly used everywhere, $ {({a^b})^c} = {a^{b \cdot c}} $ .
We can simplify this by using the exponent rule $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ .
As in the given question we also have an inverse power, there is also another rule we have to apply which is $ {a^{ - 1}} = \dfrac{1}{a} $ .
To solve exponential equations with base, use the property of power of exponential functions.
Now we will apply all the above rules and it can be written as follows: $ ( - {2^3}{x^{2 \cdot 3}})\left( {\dfrac{1}{{4{x^3}}}} \right) $ , by multiplying the powers we have, $ ( - 8{x^6})\left( {\dfrac{1}{{4{x^3}}}} \right) $ .
It can be written in fractions i.e. $ \dfrac{{ - 8{x^6}}}{{4{x^3}}} $ .
Here the division rule of division exponential is used: $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ .
So it can be written as $ - 2{x^{6 - 3}} $ . It gives the value $ - 2{x^3} $

Hence the correct answer of the exponential form is $ - 2{x^3} $.

Note: We know that exponential equations are equations in which variables occur as exponents. The formula applied before is true for all real values of $ m $ and $ n $ . We should solve this kind of problem by using the properties of exponents to simplify the problem. We have to keep in mind that if there is a negative value in the power or exponent then it will reverse the number .i.e. $ {m^{ - x}} $ will always be equal to $ \dfrac{1}{{{m^x}}} $ . We should know that the most commonly used exponential function base is the transcendental number which is denoted by $ e $ .