Problem 65 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\frac{2.24 \times 10^{4}}{5.6 \times 10^{7}}\)

Step-by-Step Solution

Answer

Scientific notation: \( 4.0 \times 10^{-4} \); Standard notation: 0.0004.

Step 1: Divide the Coefficients

Start by dividing the coefficients. In this problem, divide 2.24 by 5.6.\[ \frac{2.24}{5.6} = 0.4 \]

Step 2: Subtract the Exponents

Next, subtract the exponent of the denominator from the exponent of the numerator. The exponents are 4 and 7, respectively.\[ 4 - 7 = -3 \]

Step 3: Write in Scientific Notation

Combine the result from Step 1 with the result from Step 2 to express the answer in scientific notation.\[ 0.4 \times 10^{-3} \]

Step 4: Convert to Standard Notation

Convert the scientific notation into standard notation by moving the decimal place 3 places to the left (due to the negative exponent).\[ 0.4 \times 10^{-3} = 0.0004 \]

Key Concepts

Exponents

Exponents are powerful tools in mathematics that simplify expressing very large or very small numbers. When using scientific notation, an exponent tells us how many times the base, 10, must be multiplied or divided by itself.
In our problem, we deal with the powers of 10, specifically \[ 10^4 \] and \[ 10^7 \]. These numbers are essential because they guide us on where to move the decimal point, which is crucial, especially in scientific and engineering calculations.
- **Subtracting Exponents:** When dividing numbers in scientific notation, the rule is to subtract the exponents: \[ 10^4 \] divided by \[ 10^7 \] becomes \[ 10^{4-7} \] which is \[ 10^{-3} \]. This subtraction tells us that we must move the decimal point to the left. - **Negative Exponents:** The negative exponent here \[ 10^{-3} \] means that the decimal moves three places to the left. This is how exponents not only simplify calculations but guide the transformation of scientific to standard notation.

Coefficients

Coefficients in scientific notation are the numbers in front of the power of 10. They are important because they represent the significant figures of the value.
For the exercise provided, the coefficients are 2.24 and 5.6. We handle them separately from the exponents for clarity and ease during calculations.
- **Dividing Coefficients:** To begin solving, you divide these coefficients just like any regular numbers: \[ \frac{2.24}{5.6} = 0.4 \]. This result is crucial because it forms part of the answer you will recombine with the exponent. - **Significance of Coefficients:** In scientific notation, only one non-zero digit is usually placed before the decimal point. Therefore, the division already ensures the coefficient (0.4 in the result) stays within the range needed for proper scientific notation.
Coefficients thus help maintain accuracy and precision while working with vast variations in magnitude.

Standard Notation

Standard notation is the typical way of writing numbers without using exponents. It represents numbers in their full magnitude as opposed to their compact form in scientific notation.
To convert a number from scientific to standard notation, one has to adjust the decimal point based on the exponent. Here, we have \[ 0.4 \times 10^{-3} \]. The negative \(-3\) exponent means that we will move the decimal point three places to the left. - **Converting Example:** In our case, it transforms the number \[ 0.4 \] to \[ 0.0004 \] in standard notation. - **Understanding Conversion:** Each move of the decimal by one unit signifies a multiplication or division by 10, depending on the direction. Negative exponents mean division (hence moving left), whereas positive exponents mean multiplication (moving right).
This conversion makes interpreting real-world quantities more relatable and understandable, allowing broader accessibility for non-specialists.