Problem 50 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation without using a calculator. Write the result in scientific notation. $$ \frac{8 \times 10^{-3}}{4 \times 10^{-5}} $$

Step-by-Step Solution

Answer

The result after performing the indicated operation in scientific notation is \(2 \times 10^{2}\)

Step 1: Separate Coefficients and Powers of 10

Separate the division problem into two: the coefficients and the powers of 10. So\[\frac{8 \times 10^{-3}}{4 \times 10^{-5}}= \frac{8}{4} \times \frac{10^{-3}}{10^{-5}}\]

Step 2: Perform Division of Coefficients

Divide the coefficients \(8/4\) to get the result: \[ \frac{8}{4}= 2 \]

Step 3: Use Exponent Rules to Simplify the Powers of 10

When dividing with the same base, subtract the exponents. \(10^{-3}/10^{-5}\) becomes \(10^{(-3) - (-5)}\) or \(10^{2}\).

Step 4: Combine Results

Multiply the result of step 2 and step 3 to give the final answer: \(2 \times 10^{2}\)

Key Concepts

Division of Coefficients

When you come across a problem that requires division in scientific notation, the first thing to do is to handle the coefficients separately. Coefficients are the numbers that are multiplied by the power of ten. - In the given exercise, we have the coefficients 8 and 4. - To divide the coefficients, simply perform the arithmetic operation: divide 8 by 4. - The result of this division is 2, which becomes the coefficient of our final answer. Dealing with coefficients on their own can simplify the process since it separates numerical calculations from those involving exponents.

Powers of Ten

In scientific notation, numbers are expressed as a product of a coefficient and a power of ten. When dividing numbers in this form, the powers of ten require special attention.- In our exercise, the powers of ten are represented as \(10^{-3}\) and \(10^{-5}\).- The goal is to divide these powers, which involves exponential rules.Remember, each power of ten holds crucial information about the scale of the number, and we handle them using exponent rules to keep things organized and straightforward.

Exponent Rules

Exponent rules are the laws that govern the operations on powers, which can simplify calculations involving similar bases.- To divide expressions with the same base, subtract the exponents: \(\frac{10^{-3}}{10^{-5}}\).- Here, you subtract the exponent -5 from -3, giving you 2 as the exponent.This subtraction returns the simplified power of ten: \(10^2\). Combining this with the result from dividing the coefficients (which was 2), you get the answer in scientific notation: \(2 \times 10^2\).Understanding exponent rules helps immensely not just in division, but in other operations involving powers as well.