Problem 59 - Algebra 1: Concepts and Skills

Question

Use a calculator to perform the indicated operation. Write the result in scientific notation and in decimal form. $$ \frac{6.45 \times 10^{-6}}{4.3 \times 10^{5}} $$

Step-by-Step Solution

Answer

The result in scientific notation is \(1.50 \times 10^{-11}\) and in decimal form, it is 0.0000000000150.

Step 1: Break down the division

First, this division can be broken down into separate divisions of the coefficients and the powers of 10: \(\frac{6.45}{4.3} \times \frac{10^{-6}}{10^{5}}\).

Step 2: Perform the division of the coefficients

Divide the coefficients: \(\frac{6.45}{4.3} \approx 1.50\). Use a calculator if needed. This becomes the new coefficient.

Step 3: Reduce the powers of 10

When dividing with the same base, we subtract the exponents: \(-6 - 5 = -11. So, combine the new coefficient with the reduced power of 10 to get the result in scientific notation: \(1.50 \times 10^{-11}\).

Step 4: Express the result in decimal form

To convert scientific notation to decimal notation, move the decimal point 11 places to the left, because the exponent is negative. This gives the decimal result: 0.0000000000150.

Key Concepts

Division of Coefficients

When tackling equations in scientific notation, the first step involves dividing the coefficients. Coefficients are the tangible numbers that exist outside the scientific notation's base 10 component.
In our example, the coefficients are 6.45 and 4.3.
To divide these coefficients, simply perform regular division:

• Divide 6.45 by 4.3.
• Utilize a calculator if needed to find the approximate result, which is 1.50 in this case.

This step simplifies the numerical part of the expression, and turns it into a simpler form. The goal here is to ensure that the number stays as a manageable decimal.

The new coefficient for the scientific notation is 1.50. Always ensure to check your calculations for accuracy to maintain precision in subsequent steps.

Powers of 10

The power of 10, a key feature of scientific notation, represents the scale of the number.
When dividing expressions like these, we focus on dealing with the exponents.
In our case, we have the expression \( \frac{10^{-6}}{10^{5}} \).

• Remember, when dividing powers with the same base, you subtract the exponents.
• The operation translates to \(-6 - 5\). Calculating this gives an exponent of \(-11\).
• This means the power of 10 in the scientific notation will be \(10^{-11}\).

Getting comfortable with handling exponents is critical, as it allows us to smoothly manipulate scientific notation as a whole and prepare it for conversion into other forms.

Decimal Form

Converting scientific notation to decimal form involves a fundamental understanding of moving the decimal point.
Our scientific notation result is \(1.50 \times 10^{-11}\). The negative exponent tells us to shift the decimal point to the left.
Here's how to make the conversion:

• Identify that the exponent is \(-11\), meaning you move the decimal 11 places to the left.
• This requires adding additional zeros to accommodate the new position of the decimal point.
• Once the decimal is moved, you'll have the final decimal form: 0.0000000000150.

Paying close attention to the placement of the decimal ensures accuracy in your answer, allowing you to accurately represent very small numbers in a readable format.