Question

Using the law of exponents, simplify and write the answer in exponential form:
${{3}^{2}}\times {{3}^{4}}\times {{3}^{8}}$

Answer 1

Hint: The power or indices of a number is known as Exponents. There are various laws of exponents that are used to solve different types of sums. For this question, the law of exponent product with same base is used that is ${{\text{a}}^{m}}\times {{\text{a}}^{n}}={{\text{a}}^{m+n}}$.

Formula used:
${{\text{a}}^{m}}\times {{\text{a}}^{n}}={{\text{a}}^{m+n}}$, product with same base.

Complete step-by-step answer:
In the following question, there are three bases present. Thus, the formula of law of exponent becomes ${{\text{a}}^{m}}\times {{\text{a}}^{n}} \times {{\text{a}}^{p}}={{\text{a}}^{m+n+p}}$
Here, $m=2$, $n=4$, $p=8$ and $\text{a = 3}$.
Now, we have all values, put them in the formula ${{\text{a}}^{m}}\times {{\text{a}}^{n}} \times {{\text{a}}^{p}}={{\text{a}}^{m+n+p}}$.
Thus,
$\begin{align}
  & {{3}^{2}}\times {{3}^{4}} \times {{3}^{8}}={{3}^{2+4+8}} \\
 & ={{3}^{14}}
\end{align}$
Hence, the exponential form is ${{3}^{14}}$.

Additional information:
The various laws of exponential are explained below:
1.Product with same base: When multiplying like bases i.e. same bases with different exponents, the base remains the same and addition of exponents take place. The formula is as follows: ${{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}$.
2.Power to a power: The multiplication of powers takes place when we raise a base with a power to another power, while the base remains the same. The formula used is ${{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}}$.
3.Quotient with the same base: When the like bases are divided, the base remains the same and subtraction of exponents takes place, the denominator is subtracted from the numerator. The formula is given by $\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$.
4.Product to a power: Distribute the power to each factor, when a product is raised to a power. The formula is ${{\left( xy \right)}^{a}}={{x}^{a}}{{y}^{a}}$.
5.Zero power: Anything raised to zero is always equal to one i.e. ${{x}^{0}}=1$

Note: The conceptual knowledge about exponents and laws of exponents is required. Students should always keep in mind various laws of exponents in order to solve these types of questions. Mistakes can be by students while applying law of exponent.