Question

How do you simplify $\left( {{e}^{x}} \right)\left( {{e}^{2}} \right)?$

Answer 1

Hint: (1) For the two exponents which are the same base, as per the law we have the power of exponents.
For example: ${{a}^{n}}.{{a}^{m}}={{a}^{n+m}}$
(2) Apply the ‘rule’ of exponent.
Which is ‘add’ exponent.
Add the coefficient of the variable leaving the exponent unchanged.
Add the powers, both the variables and exponents of the variable must be the same.

Complete Step by Step solution:
We know that given question is.
$\Rightarrow \left( {{e}^{x}} \right)\left( {{e}^{2}} \right)$
Here applying the law of exponents.
i.e. ${{a}^{n}}.{{a}^{m}}={{a}^{n+m}}$
So by applying the law of exponents.
$\Rightarrow {{e}^{x}}\times {{e}^{2}}$
Can be expressed as by applying law.
$\Rightarrow {{e}^{x+2}}$
Simplified form of $\left( {{e}^{x}} \right)\left( {{e}^{2}} \right)$ is ${{e}^{x+2}}$

Additional Information:
(1) The base which is raised to the power of $'n'$ equal to the multiplication of a $n$ times.
${{a}^{n}}=a\times a\times a\times ...\times {{a}^{n}}$
Where, $a$ is the base and $n$ is the exponent.
For example: ${{3}^{1}}=3$
${{3}^{4}}=81$
$21$ there are no. of rules
(i) Product rule: ${{a}^{n}}.{{a}^{m}}={{a}^{n+m}}$
${{a}^{n}}.{{b}^{n}}={{\left( a.b \right)}^{n}}$
(ii) Quotient Rule: $\dfrac{{{a}^{n}}}{{{a}^{m}}}={{a}^{n-m}}$
$\dfrac{{{a}^{n}}}{{{b}^{n}}}={{\left( \dfrac{a}{b} \right)}^{n}}$
(iii) Power rule I: \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{n-m}}\]
Power rule II: ${{a}^{{{n}^{m}}}}{{a}^{\left( {{n}^{m}} \right)}}$
Example:
${{\left( {{2}^{3}} \right)}^{2}}={{2}^{3.2}}={{2}^{6}}=64$

Note:
(1) Identify which rule of exponent should apply to the given question, identify the base and exponent before applying the rule.
(2) While applying the product rule with the same base, keep the base as it is and add the exponents. And write the base only one time because it is the same in both terms.
(3) You can also use the different types of exponents rule in the problems as per the requirement For example: means if the numbers are in division, their base are same but the power of both number are different so here you can use the quotient rule of exponent by subtracting the power with each other.