Problem 32 - Elementary Algebra

Question

\(\frac{146 \times 10^{8}}{2 \times 10^{5}}\)

Step-by-Step Solution

Answer

7.3 \times 10^{4}

Step 1: Write the expression

Write the given expression as \( \frac{146 \times 10^{8}}{2 \times 10^{5}} \).

Step 2: Separate the constants and the powers of 10

Rewrite the expression by separating the numerical coefficients and the powers of 10: \( \frac{146}{2} \times \frac{10^{8}}{10^{5}} \).

Step 3: Simplify the constants

Divide 146 by 2 to get 73: \( 146 \div 2 = 73 \). So, the expression is now \( 73 \times \frac{10^{8}}{10^{5}} \).

Step 4: Simplify the powers of 10

Subtract the exponents of 10 as follows: \( 10^{8} \div 10^{5} = 10^{3} \). So, the expression becomes \( 73 \times 10^{3} \).

Step 5: Combine the simplified constants and powers

Combine the simplified terms to get the final result: \( 73 \times 10^{3} = 7.3 \times 10^{4} \).

Key Concepts

Scientific Notation

Scientific notation is a handy way to express very large or very small numbers. It represents numbers as a product of a numerical coefficient and a power of 10.
For example, instead of writing 500,000, you can write it as \( 5 \times 10^5 \).
This method is efficient in simplifying calculations and making sense of large values.

To convert a regular number into scientific notation, follow these steps:

• Move the decimal point in the number until you have a coefficient between 1 and 10.


• Count how many places you moved the decimal point; that number becomes your exponent of 10.


• If you moved the decimal to the left, the exponent is positive. If you moved it to the right, the exponent is negative.

For instance, the number 6,500 can be written as \(6.5 \times 10^3\). Here, 6.5 is the numerical coefficient, and the exponent of 3 indicates the decimal point has been moved 3 places to the left.

Division of Exponents

Division of exponents is simpler than it seems. When dividing numbers in scientific notation that have the same base, you can subtract the exponent of the denominator from the exponent of the numerator.
This is illustrated by the formula:
\[ \frac{{a \times 10^m}}{{b \times 10^n}} = \frac{a}{b} \times 10^{m-n} \]
Take the provided exercise for example:
\( 10^8 \) divided by \( 10^5 \) results in \( 10^{8-5} \), which is \( 10^3 \).
Here’s another way to see it:

• Remember the rule that \( \frac{{10^a}}{{10^b}} = 10^{a-b} \).


• In our case, \( \frac{{10^8}}{{10^5}} = 10^3 \).

This concept not only applies to powers of 10 but also to any base where the bases are the same.

Numerical Coefficients

Numerical coefficients are the numbers that multiply variables or units in an expression. They are essential when working with scientific notation and exponent rules.
For the given exercise, let’s focus on simplifying the numerical coefficients:
1. Initially, the expression is \( \frac{146 \times 10^8}{2 \times 10^5} \).2. Separate the constants from the powers, so it becomes \( \frac{146}{2} \times \frac{10^8}{10^5} \).3. Simplify the constants: \( \frac{146}{2} = 73 \). This gives us an intermediate result of \( 73 \times 10^3 \).4. Converting \( 73 \times 10^3 \) into a more standard scientific notation form: \( 7.3 \times 10^4 \). While doing these calculations:

• Always perform the division between numerical coefficients first.


• Then handle the exponents separately.

This ensures the expression is simplified correctly and efficiently!