Question

How do you simplify $\left( {3{y^2}} \right)\left( {2{y^3}} \right)$?

Answer 1

Hint: This problem deals with solving the expressions with exponents and bases. An expression that represents repeated multiplication of the same factor is called a power. The variable $y$ is called the base, and the number of its power is called the exponent, on the left hand side of the given equation. The exponent corresponds to the number of times the base is used as a factor.
Here some basic rule of exponents and bases are used here such as:
$ \Rightarrow {a^m} \cdot {a^n} = {a^{m + n}}$
$ \Rightarrow \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$

Complete step by step answer:
The given expression is $\left( {3{y^2}} \right)\left( {2{y^3}} \right)$, which is considered below:
$ \Rightarrow \left( {3{y^2}} \right)\left( {2{y^3}} \right)$
Now consider the first term expression, as given below:
$ \Rightarrow 3{y^2}$
Here the base is $y$ and the exponent is $2$.
Now considering the second term expression of the given expression, as given below:
$ \Rightarrow 2{y^3}$
Here the base is $y$ and the exponent is $3$.
Now multiplying the two terms of expressions as shown below:
$ \Rightarrow \left( {3{y^2}} \right)\left( {2{y^3}} \right)$
$ \Rightarrow 6{y^2} \cdot {y^3}$
Here applying the basic rule of the exponents and the bases ${a^m} \cdot {a^n} = {a^{m + n}}$, to the two terms, as shown:
$ \Rightarrow 6{y^{2 + 3}}$
$ \Rightarrow 6{y^5}$

So the simplification of the given expression $\left( {3{y^2}} \right)\left( {2{y^3}} \right)$ is given below:
$\therefore \left( {3{y^2}} \right)\left( {2{y^3}} \right) = 6{y^5}$.

Note: Please note that usually a power is represented with a base and an exponent. The base tells what number is being multiplied. The exponent, a small number written above and to the right of the base number, tells how many times the base number is being multiplied. The product rule says that to multiply two exponents with the same base, you keep the base and add the powers.
$ \Rightarrow {a^m} \cdot {a^m} = {a^{m + n}}$
$ \Rightarrow {\left( {{a^m}} \right)^n} = {a^{mn}}$
$ \Rightarrow {a^m} = {a^n}$, if the bases are the same, then the exponents have to be the same.