Question

What is 3 to the fourth power times 3 to the fifth power?

Answer 1

Hint: Here we are asked to simplify the exponents. Firstly, we will write the question into numerical form. And then we use the multiplication rule of exponents which is given by \[{{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}\]. we substitute the values in this rule and find out the required solution, which will be in the simplest form.

Complete step by step solution:
Here we are given the question in the statement form,
i.e., 3 to the fourth power times 3 to the fifth power,
which we can write in the exponential form \[{{\left( 3 \right)}^{4}}\cdot {{\left( 3 \right)}^{5}}.......\left( 1 \right)\]
we are asked to solve the above exponential expression given in the equation (1).
i.e., we need to simplify the given expression and obtain the required answer in the simplest form. Observe the given expression carefully.
We can see there are two parentheses present. In both of the parentheses, we can see that bases are equal.
Hence, the equation (1) can be written as
\[\Rightarrow {{\left( 3 \right)}^{4}}\cdot {{\left( 3 \right)}^{5}}\]
Note that the exponents inside the parenthesis are of the form \[{{a}^{m}}\cdot {{a}^{n}}\]
We have the multiplication rule of exponents which is given by,
 \[{{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}\]
Here \[m=4\]and \[n=5\].
Hence applying the multiplication rule of exponents in the parenthesis, we get
\[\Rightarrow {{\left( 3 \right)}^{4}}\cdot {{\left( 3 \right)}^{5}}={{\left( 3 \right)}^{4+5}}\]
We know that \[4+5=9\]
Hence the above expression becomes,
 \[\Rightarrow {{\left( 3 \right)}^{4}}{{\left( 3 \right)}^{5}}={{\left( 3 \right)}^{9}}\].
Hence, the simplest form of \[{{\left( 3 \right)}^{4}}{{\left( 3 \right)}^{5}}\]is given by \[{{3}^{9}}\]or \[19683\]

So, the 3 to the fourth power times 3 to the fifth power is \[{{3}^{9}}\]or \[19683\].

Note:
Don’t be confused between the exponents and powers. Students must remember the rule of exponents to simplify such problems. We need to be careful while applying the rules. It's necessary to use the correct rule to split the terms and simplify the answer.
The exponential rules are:
1. Multiplication rule: \[{{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}\]
2. Division rule: \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
3. Power of a power rule: \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]
4. Power of a product rule: \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]
5. Power of a fraction rule: \[{{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}\]
6. Zero exponent: \[{{a}^{0}}=1\]
7. Negative exponents: \[{{a}^{-x}}=\dfrac{1}{{{a}^{x}}}\]