Question

How do you simplify ${3^n}{.3^{2n}}$ ?

Answer 1

Hint: We are given the product of two exponential functions ${3^n}$ and ${3^{2n}}$ in this question. 3 is raised to the power n and 2n, so 3 is called the base, and n and 2n are called its powers. When a number is raised to the power “n”, it means that the number is multiplied with itself “n” times, for example, let ${a^n}$ be an exponential function, it means that the number “a” is multiplied with itself “n” times. So, ${3^n}$ means that 3 is multiplied with itself n times and ${3^{2n}}$ means that 3 is multiplied with itself 2n times. This way we can find the value of any number that is raised to the power of some other number. Applying the law of exponents, we can simplify the given expression.

Complete step-by-step solution:
We have to simplify ${3^n}{.3^{2n}}$
We know that ${a^x}.{a^y} = {a^{x + y}}$ , so we get –
$
  \Rightarrow {3^n}{.3^{2n}} = {3^{n + 2n}} \\
   \Rightarrow {3^n}{.3^{2n}} = {3^{3n}} \\
 $
We also know that ${a^{xy}} = {({a^x})^y}$ , so we get –
${3^n}{.3^{2n}} = {({3^3})^n}$
${3^3}$ means that 3 is multiplied with itself 3 times, so ${3^3} = 3 \times 3 \times 3 = 27$
$ \Rightarrow {3^n}{.3^{2n}} = {27^n}$
Hence the simplified form of ${3^n}{.3^{2n}}$ is ${27^n}$ .

Note: We have used the law of exponent which states that when a number raised to some power is multiplied with that number raised to some other power then keeping the base same we add the powers, that is, ${a^x}.{a^y} = {a^{x + y}}$ . We have used one more law of exponent which states that when a number raised to some power is again raised to another power then keeping the base same, we multiply the powers and vice-versa, that is, ${({a^x})^y} = {a^{x \times y}}$ .