Problem 101 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$\frac{2.4 \times 10^{-2}}{4.8 \times 10^{-6}}$$

Step-by-Step Solution

Answer

The result of \(\frac{2.4 \times 10^{-2}}{4.8 \times 10^{-6}}\) is \(5 \times 10^{3}\)

Step 1: Division of Coefficients

The first step involves dividing the coefficients of the given scientific numbers. This means dividing 2.4 by 4.8, which is equal to 0.5.

Step 2: Subtraction of Exponents

Subtract the exponent of the denominator from the exponent of the numerator. This means we subtract -6 from -2, which equals to 4 (-2 - (-6) = 4).

Step 3: Combine Coefficients and Exponents

The final scientific number is formed by combining the coefficient (from step 1) with the exponent (from step 2) to yield \(0.5 \times 10^{4}\).

Step 4: Convert into standard form

To express the number in standard scientific notation, we adjust the coefficient to be a value between 1 and 10 and adjust the exponent accordingly. This yields \(5 \times 10^{3}\).

Key Concepts

Division of Coefficients

In the realm of scientific notation, dealing with coefficients might seem tricky at first, but it's quite straightforward. Here is how it works: when dividing numbers in scientific notation, you start by dividing the coefficients.
In our exercise, we have to divide 2.4 by 4.8. Always remember, the coefficient is the number at the front of the scientific notation, like 2.4 in our case.
To perform the division, you simply calculate:

• 2.4 ÷ 4.8 = 0.5

The result of your division will be the new coefficient that will factor into the subsequent calculations.

Subtraction of Exponents

Handling exponents during division in scientific notation involves subtraction. For every division operation, the process is simple: subtract the exponent in the denominator from that in the numerator.
Looking at our example, we have exponents

• Numerator exponent: -2
• Denominator exponent: -6

To find the resulting exponent, we calculate

• -2 - (-6) = 4

Here, subtraction of a negative number is like adding its positive counterpart, hence the final result is 4.

Scientific Notation Conversion

After dividing the coefficients and subtracting the exponents, you end up with a number that is not yet formatted as a standard scientific notation.
It's crucial to properly combine these operations:
We have a coefficient of 0.5 and an exponent of 4, resulting in the expression:

• 0.5 × 10⁴

This combination forms the expression, but it often needs fine-tuning for it to be accepted as standard scientific notation, hence the next step.

Standard Scientific Notation

To ensure that a number is in standard scientific notation, the coefficient must be between 1 and 10. If it is not, you must adjust it accordingly and modify the exponent.
In our example, the coefficient 0.5 is below the required range. Thus, we multiply 0.5 by 10 to correct the coefficient, which becomes 5.
The multiplication by 10 suggests the need to adjust the exponent by subtracting 1, giving us:

• 5 × 10³

This expression is now in standard scientific notation, which clearly communicates both magnitude and precision.