Problem 76 - Beginning and Intermediate Algebra

Question

Write the answer using scientific notation. $$\frac{1.1 \times 10^{-40}}{2.0 \times 10^{-71}}$$

Step-by-Step Solution

Answer

The short answer is: \(5.5 \times 10^{30}\).

Step 1: Identify the decimal and exponent parts

We're given the expression: \(\frac{1.1 \times 10^{-40}}{2.0 \times 10^{-71}}\) Here, the decimal parts are 1.1 and 2.0; the exponents are -40 and -71.

Step 2: Divide the decimal parts

Divide the decimal parts (1.1 and 2.0) of the numbers: \(\frac{1.1}{2.0} = 0.55\)

Step 3: Subtract the exponents

Subtract the exponents (-40 and -71) of the numbers: \((-40) - (-71) = -40 + 71 = 31\)

Step 4: Combine the results

Now, combine the results obtained in steps 2 and 3. \(0.55 \times 10^{31}\) The answer is written in scientific notation as: \(5.5 \times 10^{30}\)

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with scientific notation. Exponents allow for the representation of a number as a base raised to a power. These rules simplify calculations and help manage very large or very small numbers efficiently.

• When multiplying numbers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• For division, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising a power to a power means multiplying the exponents: \((a^m)^n = a^{m\times n}\).

In the exercise given, you use the division rule. This involves subtracting exponents, as was demonstrated with \((-40) - (-71) = 31\). This helps transform the expression to a simplified form in scientific notation.

Decimal Operations

Decimal operations are essential when dealing with scientific notation. Scientific notation separates numbers into decimal and exponential components. In such expressions, performing operations on decimals requires precision and sometimes can lead to rounding. For your exercise:

• Identify the decimal components: in this case, 1.1 and 2.0.
• Execute basic arithmetic operations like addition, subtraction, multiplication, or division.
• Ensuring accuracy, especially in division, is key to maintaining integrity in calculations.

The exercise demonstrates dividing \(1.1\) by \(2.0\) resulting in \(0.55\). This calculation is straightforward due to the simplicity of the numbers, but practice makes handling more complex numbers easy.

Scientific Calculations

Scientific calculations often involve working with very small or large numbers in a compact form. This is where scientific notation shines.

• It expresses numbers as a product of a decimal and a power of ten.
• Order of operations is essential to ensure correct calculations.
• Reformatting numbers into scientific notation can make complex calculations more manageable.

In the given exercise, you're combining the results from calculated decimals and exponents: \(0.55 \times 10^{31}\). For it to fit proper scientific notation, numbers should typically have a single non-zero digit before the decimal. Adjust to \(5.5 \times 10^{30}\), ensuring the result remains mathematically accurate while adhering to the convention of scientific notation.