Problem 62 - College Algebra with Modeling and Visualization

Question

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \left(9.87 \times 10^{6}\right)\left(34 \times 10^{11}\right) $$

Step-by-Step Solution

Answer

The expression approximates to \(3.36 \times 10^{19}\).

Step 1: Understand Scientific Notation Multiplication

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents of the powers of 10.

Step 2: Multiply the Coefficients

Let's multiply the numerical coefficients: \(9.87 \times 34 = 335.58\).

Step 3: Add the Exponents

The exponents of the powers of 10 are 6 and 11. Add these exponents together: \(6 + 11 = 17\).

Step 4: Combine the Results

Combine the multiplied coefficient and the sum of the exponents: The result is \(335.58 \times 10^{17}\).

Step 5: Convert to Standard Scientific Notation

To convert \(335.58 \times 10^{17}\) into standard scientific notation, adjust the coefficient to fall between 1 and 10. Move the decimal two places to the left, changing \(335.58\) to \(3.3558\), and increase the exponent by 2: The final result is \(3.3558 \times 10^{19}\).

Step 6: Final Approximation Using a Calculator

Confirm the result using a calculator. The multiplication verifies the scientific notation conversion: The calculation confirms it as approximately \(3.36 \times 10^{19}\), adjusting to three significant figures.

Key Concepts

Scientific Notation Multiplication

Scientific notation is a convenient way to handle both very large and very small numbers by expressing them as a product of a number between 1 and 10 and a power of ten. This is especially helpful for calculations involving astronomical sizes or tiny atomic measurements. When multiplying numbers in scientific notation:

• Multiply the coefficients, which are the decimal numbers, not the powers of ten.
• Add the exponents of the powers of 10 together.

For example, in the expression \((9.87 \times 10^{6})(34 \times 10^{11})\), you first multiply 9.87 by 34 to get 335.58. Then, you handle the powers of ten by adding the exponents, 6 and 11, resulting in a sum of 17. So the initial answer is \(335.58 \times 10^{17}\). This process simplifies dealing with very large numbers, making calculations feasible and easier to understand.

Exponents Addition

Adding exponents is a key step when multiplying numbers in scientific notation. This operation is conceptually simple but crucial for combining the powers of ten correctly. Exponents indicate how many times a number (the base) is multiplied by itself. For multiplication of powers of ten:

• If you have \(10^a \times 10^b\), you add the exponents: \(a+b\).

This rule emerges because \(10^a\) represents a number with 'a' number of zeros following a 1, and the same applies to \(10^b\). Thus, the total zeros become the sum of a and b. Using this principle for \(10^6\) and \(10^{11}\), their product is \(10^{17}\), combining the growth of two large numbers efficiently.

Standard Scientific Notation Conversion

After performing multiplications in scientific notation, it is important to convert your result to standard scientific notation. This form has the coefficient between 1 and 10. This makes it easy to compare and use the results in further calculations, especially in scientific contexts.To convert into standard scientific notation:

• Adjust the coefficient so that it lies between 1 and 10. This often involves moving the decimal point.
• Each shift to the left for the decimal increases the exponent by 1; conversely, each shift to the right decreases it by 1.

In our example, the number \(335.58 \times 10^{17}\) is converted by moving the decimal point two places left to become \(3.3558\). Correspondingly, you increase the exponent by 2, resulting in \(3.3558 \times 10^{19}\). Standardizing this form ensures the results are understandable and neatly fits scientific standards.