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Question: How do you simplify \(2{e^{ - 2x}}\;{\text{and}}\;2{e^{ - x}}?\)...

How do you simplify 2e−2x  and  2e−x?2{e^{ - 2x}}\;{\text{and}}\;2{e^{ - x}}?

Explanation

Solution

To simplify the given terms, use law of indices for negative power. Since the terms only contain exponentials and in the simplest form of an exponential there should be no negative power or no zero or one power. So simplify the given terms accordingly.Law of indices for negative power is given as x−a=1xa{x^{ - a}} = \dfrac{1}{{{x^a}}}.Use this law to simplify the given terms.

Complete step by step solution:
In order to simplify the given terms 2e−2x  and  2e−x2{e^{ - 2x}}\;{\text{and}}\;2{e^{ - x}}, we first check what complex operations does the terms are consist of. On seeing both terms we can see that there is only exponential function in the given terms, and also we know that an exponential function in its simplest form does not have any negative power, zero power and one power.

But in the given terms 2e−2x  and  2e−x2{e^{ - 2x}}\;{\text{and}}\;2{e^{ - x}} we can see that there’s negative power present, so we will remove that negative power with help of law of indices for negative powers, which is given as a term with positive power is the multiplicative inverse of the same term with negative power. Mathematically it can be expressed as
x−a=1xa{x^{ - a}} = \dfrac{1}{{{x^a}}}
Using this, we can write
2e−2x=12e2x  and  2e−x=12ex2{e^{ - 2x}} = \dfrac{1}{{2{e^{2x}}}}\;{\text{and}}\;2{e^{ - x}}= \dfrac{1}{{2{e^x}}}
Therefore these are the simplified terms.

Note: When simplifying this type of terms, you have to first check how complex the term is and also what are the functions it consists of then simplify the term. If the term contains more than one function, then use the BODMAS rule for the preference and then simplify the term accordingly.