Question

How do you simplify $\left( {{x^3}} \right)\left( {4{x^5}} \right)$?

Answer 1

Hint:
Here we are asked to simplify the exponents. Firstly, we will take out the constant term out of the parenthesis. And collect the exponents in one parenthesis. 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 obtain the required solution, which will be in the simplest form.

Complete step by step solution:
Here we are given the exponents of the form $\left( {{x^3}} \right)\left( {4{x^5}} \right)$ …… (1)
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 result in the simplest form.
Observe the given expression carefully.
We see that among there are two parenthesis present. In one of the parenthesis there is a constant term present which is multiplied to an exponent.
Firstly, we will take out the constant term 4 out of the parenthesis.
Hence the equation (1) becomes,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = \left( {{x^3}} \right)4 \cdot \left( {{x^5}} \right)$
This can also be written as,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = 4\left( {{x^3}} \right)\left( {{x^5}} \right)$
Now collect the exponents present in one parenthesis. So that it becomes easier to solve and simplify.
Hence we get,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = 4\left( {{x^3} \cdot {x^5}} \right)$
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 $m = 3$ and $n = 5$.
Hence applying multiplication rule of exponents inside the parenthesis, we get,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = 4\left( {{x^{3 + 5}}} \right)$
We know that $3 + 5 = 8$.
Hence the above expression becomes,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = 4\left( {{x^8}} \right)$
This can also be written without the parenthesis. Hence we get,
$ \Rightarrow \left( {{x^3}} \right)\left( {4{x^5}} \right) = 4{x^8}$

Hence the simplest form of $\left( {{x^3}} \right)\left( {4{x^5}} \right)$ is given by $4{x^8}$.

Note:
Students must remember the rules 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 rules of exponents are given below.
(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 : ${({a^m})^n} = {a^{mn}}$
(4) Power of a product rule : ${(ab)^m} = {a^m}{b^m}$
(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 exponent : ${a^{ - x}} = \dfrac{1}{{{a^x}}}$