Question

How do you simplify \[{({x^2})^4} \times 3{x^5}\] and write it using only positive numbers?

Answer 1

Hint: In the given question, we have been asked to simplify the exponent and write it using positive numbers only. In order to proceed with the following question we need to know about Exponents and rules associated with them. Exponent is basically the number of times a number is multiplied with itself. There are many rules related to exponents such as Product rule, Power rule, Quotient rule, Zero rule, etc. In this question we’ll use the power rule, which states that when we have power raised to a power, we just have to multiply the powers.

Formula used: $ {x^{{m^n}}} = {x^{m \times n}} $
 $ {x^a} \times {x^b} = {x^{a + b}} $

Complete step by step solution:
We are given,
\[ \Rightarrow {({x^2})^4} \times 3{x^5}\]
If two powers have the same base then you can multiply the powers (power rule)
 $ {x^{{m^n}}} = {x^{m \times n}} $
\[ \Rightarrow {x^8} \times 3{x^5}\]
Since, both the $ x $ have different powers and they are to be multiplied. We’ll use the basic law of exponent “Product Rule”
  $ {x^a} \times {x^b} = {x^{a + b}} $
\[ \Rightarrow 3{x^{8+5}}\]
\[ \Rightarrow 3{x^{13}}\]
So, the correct answer is “\[ 3{x^{13}}\]”.

Note: There are some special cases in exponents such as
When the exponent is $ 0 $ , the expression is always equal to $ 1 $
When the exponent is $ 1 $ , the expression is equal to the base itself, since a number multiplied by itself once is equal to the number itself.
When the exponent is negative, the result turns into a fraction.
You can have a variable as base and a number as power such as $ {a^3} $ , which basically represents $ a \times a \times a $
You can have a number as base and variable as power such as, $ {2^n} $ which basically represents that 2 has to be multiplied by itself $ n $ times.