Question

How do you simplify \[{\left( {{{10}^4}} \right)^2}?\]

Answer 1

Hint: This question involves the operation of addition/ subtraction/ multiplication/ division. Also, we need to know the basic algebraic formula with the involvement of exponent components. We need to know the process of multiplication with the exponents. We need to know how to expand the \[{10^n}\] terms. The final answer would be a most simplified form of a given question.

Complete step-by-step answer:
The given question is shown below,
 \[{\left( {{{10}^4}} \right)^2}?\] \[ \to \left( 1 \right)\]
We know, that
 \[{\left( {{x^a}} \right)^b} = {x^{a \cdot b}} \to \left( 2 \right)\]
By comparing the equation \[\left( 1 \right)\] and \[\left( 2 \right)\] , we get
 \[\left( 1 \right) \to {\left( {{{10}^4}} \right)^2} = ?\]
 \[\left( 2 \right) \to {\left( {{x^a}} \right)^b} = {x^{a \cdot b}}\]
So, the value of \[x\] is \[10\] the value of \[a\] is \[4\] and the value of \[b\] is \[2\] . So, let’s substitute these values in the equation \[\left( 2 \right)\] , we get
 \[\left( 2 \right) \to {\left( {{x^a}} \right)^b} = {x^{a \cdot b}}\]
 \[{\left( {{{10}^4}} \right)^2} = {10^{4 \times 2}} \to \left( 3 \right)\]
We know that,
 \[4 \times 2 = 8\]
So, the equation \[\left( 3 \right)\] becomes,
 \[\left( 3 \right) \to {\left( {{{10}^4}} \right)^2} = {10^{4 \times 2}}\]
 \[{\left( {{{10}^4}} \right)^2} = {10^8} \to \left( 4 \right)\]
In the above equation, we have \[{10^8}\] . It can be written as follows,
 \[{10^8} = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\]
 \[{10^8} = 100000000\]
So, the equation \[\left( 4 \right)\] becomes,
 \[{\left( {{{10}^4}} \right)^2} = {10^8} = 100000000\]
So, the final answer is,
 \[{\left( {{{10}^4}} \right)^2} = 100000000\]
So, the correct answer is “100000000”.

Note: In these types of questions we would involve the operation of addition/ subtraction/ multiplication/ division. Remember the algebraic formula with the involvement of exponent compounds. Also, note that if we have \[{10^2}\] in the question the answer will be \[100\] if we have \[{10^3}\] in the question the answer will be \[1000\] . From these examples, we can get that number we have in the power of \[10\] that number of zero would be placed after \[1\] in the final answer. So, that’s why while we have \[{10^8}\] , the final answer has eight zeroes in the final answer. Also, we can use a scientific calculator to solve these types of problems. To solve these types of questions we would compare the given question with a suitable algebraic formula.