Question

How do you simplify $ {(2{x^0}{y^2})^{ - 3}} \cdot 2y{x^3} $ and write it using only positive exponents?

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^0}{y^2}) $ is the base and the number $ - 3 $ 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 answer:
There is one basic exponential rule that is commonly used everywhere,
 $ {({a^m})^n} = {a^{m \cdot n}} $ .
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. We will first break the first part by exponential rule i.e.
 $ \Rightarrow {({x^0})^{ - 3}} = {x^{0 \times - 3}} $ .
There is one basic rule that we know that
 $ {a^0} = 1 $ .
So it gives us $ {x^0} = 1 $ . Again we have
 $ {({y^2})^{ - 3}} = {y^{2 \times - 3}} = {y^{ - 6}} $ .
Similarly with the inverse formula,
 $ \Rightarrow{2^{ - 3}} = \dfrac{1}{{{2^3}}} = \dfrac{1}{8} $ .
 Now we will apply all the above rules and it can be written as follows:
 $ {({2^{}}{x^0}{y^2})^{ - 3}} = \dfrac{{{y^{ - 6}} \times 1}}{8} $ ,
by writing all the terms all together:
 $ \Rightarrow\dfrac{{{y^{ - 6}}}}{8} \times 2y{x^3} $ .
We will now solve it:
 $ \Rightarrow\dfrac{{2 \times {y^{ - 6 + 1}} \times {x^3}}}{8} = \dfrac{{{y^{ - 5}}{x^3}}}{4} $
Hence the required answer of the exponential form is $ \dfrac{{{y^{ - 5}}{x^3}}}{4} $ .
So, the correct answer is “ $ \dfrac{{{y^{ - 5}}{x^3}}}{4} $ ”.

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 $ .