Simplify (6*10^5)^2

Solution:

Step 1:

Utilize the power of a product rule on $6\times {10}^5$. Write it as $(6{)}^2\times ({10}^5{)}^2$.

Step 2:

Square the number $6$ to get $36$. So, it becomes $36\times ({10}^5{)}^2$.

Step 3:

Handle the exponentiation of ${10}^5$ raised to the power of $2$.

Step 3.1:

Invoke the power of a power rule, which states $(a^m{)}^n=a^{mn}$. Therefore, $36\times {10}^{5\times 2}$.

Step 3.2:

Calculate $5\times 2$ to get $10$. The expression now is $36\times {10}^{10}$.

Step 4:

Adjust the number $36$ to $3.6$ and compensate by increasing the exponent of $10$ by $1$ to maintain equivalence. The expression becomes $3.6\times {10}^{11}$.

Step 5:

Present the final answer in both scientific notation and expanded form.

• Scientific Notation: $3.6\times {10}^{11}$
• Expanded Form: $360,000,000,000$

Knowledge Notes:

The problem involves simplifying an expression with a base number and an exponent. Here are the relevant knowledge points:

1. Power of a Product Rule: When a product of bases is raised to an exponent, the exponent applies to each base individually. The rule is expressed as $(ab{)}^n=a^n{b}^n$.

2. Squaring a Number: Squaring a number means raising it to the power of 2. For example, $6^2=6\times 6=36$.

3. Power of a Power Rule: When an exponent is raised to another exponent, you multiply the exponents. The rule is $(a^m{)}^n=a^{mn}$.

4. Scientific Notation: A way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written as the product of a number between 1 and 10 and a power of 10. For example, $3.6\times {10}^{11}$.

5. Expanded Form: Writing a number to show the value of each digit. It is the opposite of scientific notation, where a number is written out according to its place value. For example, $360,000,000,000$.

Understanding these concepts allows for the systematic simplification of expressions involving exponents, which is essential in various fields of mathematics and science.