Problem 89 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \left(1.6 \times 10^{15}\right)\left(4 \times 10^{-11}\right) $$

Step-by-Step Solution

Answer

The answer is \(6.4 \times 10^{4}\).

Step 1: Understanding the problem

Here there are two numbers written in scientific notation \(1.6 \times 10^{15}\) and \(4 \times 10^{-11}\), which need to be multiplied.

Step 2: Multiplication

Next, perform the multiplication separately for the decimal components (the numbers not involving the 10s) and for the \(10^a\) components. \[ (1.6 \times 4) \times (10^{15} \times 10^{-11})\]

Step 3: Perform the operations

Calculate the results. \[ (6.4) \times (10^{15-11})\] Here, subtract the exponent of the denominators from the exponent of the numerators based on the properties of exponents.

Step 4: Final Result

Simplify the above expression to get the result in scientific notation. The answer will be \[ 6.4 \times 10^{4} \] Though not required here, if any rounding was required the problem specifically states to round to two decimal places.

Key Concepts

Multiplication of Exponents

Multiplying numbers in scientific notation involves two main components: the decimal factors and the exponents of 10. In scientific notation, numbers take the form \( a \times 10^b \), where \( a \) is the decimal factor, and \( b \) is the exponent of 10. To multiply two such numbers, like \( 1.6 \times 10^{15} \) and \( 4 \times 10^{-11} \), you should first multiply the decimal factors: \( 1.6 \times 4 = 6.4 \).
Next, you apply the rules of exponents to the powers of 10. In general, when multiplying like bases, you add their exponents. So, \( 10^{15} \times 10^{-11} \) becomes \( 10^{15+(-11)} = 10^4 \).
This gives a result in scientific notation as \( 6.4 \times 10^4 \), completing the multiplication process.

Properties of Exponents

Understanding the properties of exponents is crucial when working with scientific notation. One vital property is that when you multiply numbers with the same base, like 10 in our example, you add their exponents: \( a^m \times a^n = a^{m+n} \).
This relationship simplifies the process of handling large numbers by breaking them down into smaller calculations involving the exponents and decimal factors separately.

• If dividing, you would subtract the exponent in the denominator from the exponent in the numerator.
• If raising a power to another power, then you would multiply the exponents.

For instance, \( 10^{15} \times 10^{-11} \) results in \( 10^{15 - 11} = 10^4 \). This is how we simplify complex exponential expressions into simpler forms.

Decimal Factor Rounding

When you write answers in scientific notation, sometimes you'll need to round decimal factors. Rounding ensures precision, especially when dealing with large or small numbers. In the context of scientific notation, the decimal part \( a \) is significant, and rounding helps maintain its accuracy to a given number of decimal places.
In this exercise, if rounding were necessary, we would round \( a \) to two decimal places. For example, if you end up with a result like \( 6.432 \), you would round it to \( 6.43 \).
Ensure you follow the rounding rules: if the next digit is 5 or more, round up. If it is less than 5, round down. Thus, precision is kept uniform across solutions in scientific notation.