Problem 33
Question
\(\frac{164 \times 10^{7}}{2 \times 10^{-5}}\)
Step-by-Step Solution
Verified Answer
82 \( \times \) \( 10^{12} \)
1Step 1: Simplify the expression
The expression is \(\frac{164 \times 10^{7}}{2 \times 10^{-5}}\). Write it as \(\frac{164}{2} \times \frac{10^{7}}{10^{-5}}\).
2Step 2: Divide the coefficients
Calculate \(\frac{164}{2} = 82\).
3Step 3: Apply exponent rules
Using the rule \( \frac{a^m}{a^n} = a^{m-n} \), calculate the exponent part: \( \frac{10^{7}}{10^{-5}} = 10^{7 - (-5)} = 10^{7 + 5} = 10^{12} \).
4Step 4: Combine results
Now, combine the results from Step 2 and Step 3: \( 82 \times 10^{12} \).
Key Concepts
Exponent RulesScientific NotationDivision of Coefficients
Exponent Rules
Understanding exponent rules is key when simplifying expressions involving powers.
When we divide terms that have the same base, we subtract the exponents.
For instance, \(\frac{a^m}{a^n} = a^{m-n}\).
This rule helps simplify complex expressions easily.
Remember:
When we divide terms that have the same base, we subtract the exponents.
For instance, \(\frac{a^m}{a^n} = a^{m-n}\).
This rule helps simplify complex expressions easily.
Remember:
- When multiplying terms with the same base, add the exponents: \((a^m \times a^n = a^{m+n})\).
- When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
Scientific Notation
Scientific notation is a way to express very large or very small numbers.
It is written as a product of a coefficient and a power of ten.
For example, 164 can be written as \(1.64 \times 10^2\).
This notation makes calculations easier, especially with large or small numbers.
The coefficient should be a number greater than or equal to 1 and less than 10.
In the original exercise, \[164 \times 10^7\] is already in scientific notation.
When dividing or multiplying numbers in scientific notation:
It is written as a product of a coefficient and a power of ten.
For example, 164 can be written as \(1.64 \times 10^2\).
This notation makes calculations easier, especially with large or small numbers.
The coefficient should be a number greater than or equal to 1 and less than 10.
In the original exercise, \[164 \times 10^7\] is already in scientific notation.
When dividing or multiplying numbers in scientific notation:
- Handle the coefficients separately, just like regular numbers.
- Apply exponent rules to the powers of ten.
Division of Coefficients
Dividing coefficients is a straightforward step in simplifying expressions.
Let's consider the original exercise \[ \frac{164 \times 10^7}{2 \times 10^{-5}} \].
First, divide the coefficients: \(\frac{164}{2} = 82\).
This step involves basic arithmetic and should be performed before dealing with the exponents.
Doing so reduces the expression to a simpler form.
For example, handling the coefficients separately helps avoid confusion with other parts of the expression.
After dividing the coefficients, proceed with the exponent rules.
Following these basic steps ensures a clearer and more accurate simplification process.
Let's consider the original exercise \[ \frac{164 \times 10^7}{2 \times 10^{-5}} \].
First, divide the coefficients: \(\frac{164}{2} = 82\).
This step involves basic arithmetic and should be performed before dealing with the exponents.
Doing so reduces the expression to a simpler form.
For example, handling the coefficients separately helps avoid confusion with other parts of the expression.
After dividing the coefficients, proceed with the exponent rules.
Following these basic steps ensures a clearer and more accurate simplification process.