Simplify (6*10^5)^2
The problem is asking to calculate the square of the expression (6_10^5). Squaring a number or an expression means multiplying the number or the expression by itself. Therefore, you need to multiply (6_10^5) by (6*10^5) to simplify the original expression. The process will involve the rules of exponents and multiplication to carry out the simplification.
$\left(\left(\right. 6 \cdot \left(10\right)^{5} \left.\right)\right)^{2}$
Answer
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:
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 $(ab)^{n} = a^{n}b^{n}$.
Squaring a Number: Squaring a number means raising it to the power of 2. For example, $6^2 = 6 \times 6 = 36$.
Power of a Power Rule: When an exponent is raised to another exponent, you multiply the exponents. The rule is $(a^{m})^{n} = a^{mn}$.
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, $3.6 \times 10^{11}$.
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.
