Problem 86 - Elementary Algebra

Question

\(L=\frac{9.45 \times 10^{6}}{\left(5 \times 10^{1}\right)(9.8)(2)}\)

Step-by-Step Solution

Answer

1.93 \times 10^{4}

Step 1: Identify the Given Values

Identify and note down the given values in the expression: - Numerator: \(9.45 \times 10^{6}\) - Denominator components: \(5 \times 10^{1}\), 9.8, and 2.

Step 2: Simplify the Denominator

Multiply the values in the denominator: \(d = (5 \times 10^{1}) \times 9.8 \times 2\)

Step 3: Calculate Multiplied Denominator

Compute the multiplication: \[ 5 \times 10 \times 9.8 \times 2 = 49 \times 10 = 490 \]

Step 4: Substitute and Simplify

Substitute the simplified denominator back into the equation: \[ L = \frac{9.45 \times 10^{6}}{490} \]

Step 5: Perform Division

Divide the values: \[ L = \frac{9.45 \times 10^{6}}{490} = 1.93 \times 10^{4}\]

Key Concepts

Scientific Notation

Scientific notation is a system used to express very large or very small numbers. It helps make these numbers easier to work with and understand. This notation represents numbers as a product of a number between 1 and 10 and a power of 10. For example, the number 9,450,000 can be written as \(9.45 \times 10^6\). This format is particularly useful in fields like science and engineering.

When dividing numbers in scientific notation, you follow these steps:

• Divide the coefficients (the numbers in front of the powers of 10).
• Subtract the exponents if the terms are divided.

For instance, dividing \(9.45 \times 10^6\) by \(5 \times 10^1\), you would divide 9.45 by 5 and subtract the exponents 6 and 1:

• Coefficient: \(9.45 ÷ 5 = 1.89\)
• Exponent: \(6 - 1 = 5\)

Hence, the result would be \(1.89 \times 10^5\). This method keeps calculations manageable, especially when working with very large or small values.

Fraction Simplification

Simplifying fractions is a key skill in mathematics. It involves making a fraction as simple as possible, usually by dividing the numerator and the denominator by their greatest common divisor (GCD). In our exercise, we have the fraction \(\frac{9.45 \times 10^6}{(5 \times 10^1)(9.8)(2)}\).

To simplify, we first need to handle the multiplication in the denominator. Here’s how:

• Calculate \(5 \times 10^1 = 50\)
• Multiply 50 by 9.8: \(50 \times 9.8 = 490\)
• Then, multiply the result by 2: \(490 \times 2 = 980\)

Now, we can rewrite the fraction as \(\frac{9.45 \times 10^6}{980}\).

The next step is to simplify this fraction. Since 9.45 and 980 do not have a common divisor that simplifies easily, we perform the division directly:
\(9.45 ÷ 980 = 0.009648\).
Therefore, the fraction simplifies to \(0.009648 \times 10^6\), which is essentially the value in scientific notation.

Multiplication

Multiplication is another fundamental operation in math. It is crucial for simplifying expressions and calculating values in scientific notation. To multiply two numbers, you combine their coefficients and their powers of 10 separately. For instance, if you multiply \(5 \times 10^1\) by 9.8, you follow these steps:

• Multiply the coefficients: \(5 \times 9.8 = 49\)
• Add the exponent \^1: \(10^1\) becomes \(10\), giving 49 as the intermediate product.

Then, multiply 49 by 2:
\(49 \times 2 = 98\). Summarized, the multiplication of logarithms in scientific notation involves multiplying the numeric parts and adding the exponents.
Here, since there isn’t any additional power of 10, the result remains as direct multiplication. Hence, proper handling of multiplication and exponents simplifies manipulation and understanding of large and small numbers in scientific notation.