Problem 88 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientifi c notation answer to two decimal places. $$\left(2 \times 10^{4}\right)\left(4.1 \times 10^{3}\right)$$

Step-by-Step Solution

Answer

The product of \(2 * 10^4\) and \(4.1 * 10^3\) in scientific notation is \(8.2 * 10^7\).

Step 1: Multiply the mantissas

Multiply the mantissa terms together: \(2 * 4.1\) which equals \(8.2\)

Step 2: Add the exponents

Next, add the exponent terms together: \(4 + 3\) which equals \(7\)

Step 3: Combine the results

Finally, combine the multiplied mantissas with the sum of the exponents. This gives us \(8.2 * 10^7\)

Key Concepts

Multiplication of Exponents

When handling numbers in scientific notation, it’s common to encounter multiplication of exponents. Scientific notation involves expressing numbers as a product of a number (known as the mantissa) and ten raised to a power (the exponent). In our example, we have:

• \(2 \times 10^{4}\)
• \(4.1 \times 10^{3}\)

To multiply these numbers, we must multiply the mantissas (2 and 4.1) and then deal with the exponents (4 and 3). You're essentially combining the powers of ten by working with their exponents, making the process efficient.The key principle here is that multiplying numbers in scientific notation involves adding their exponents, simplifying calculations significantly.
For instance, multiplying the numbers results in querying: \(10^{4+3}\), leading you to \(10^{7}\). This step dives into exponent theory, where adding exponents aligns with multiplication's underlying principles.

Mantissa

The mantissa is a crucial component of numbers in scientific notation, representing the decimal number before the ten raised to an exponent. It must be a number between 1 and 10. In our example multiplication, the mantissas are 2 and 4.1.
Working with the mantissa is often straightforward: simply multiply them together like basic decimal numbers. In this case:\[2 \times 4.1 = 8.2\]After performing this multiplication, we obtain a new mantissa for our scientific notation.
The mantissa makes it easy to work with very large or very small numbers, as it provides a compact form to express them in meaningful terms. Always ensure your mantissa is properly multiplied before proceeding to combine with the exponent part of scientific notation.

Rounding Decimals

When calculating mantissas, you might end up with long decimal numbers. There's often a need to round the mantissa to keep the scientific notation concise and easy to read. Typically, mantissas are rounded to two decimal places unless specific instructions are given.
Rounding is a process that simplifies decimals by eliminating less significant digits. Here's how it generally works:

• Look at the digit in the third decimal place.
• If this digit is 5 or more, add 1 to the second decimal's digit.
• If it's less, simply drop the extra digits.

In our example, the mantissa 8.2 already meets the criteria, as it doesn't extend into unnecessary decimal places. Rounding maintains the accuracy necessary for scientific work while making the numbers manageable.

Exponents Addition

In scientific notation, knowing how to perform exponents addition is crucial when dealing with multiplication. This task involves working with the powers of ten associated with each number. The concept relies on the fundamental property of exponents: when multiplying like bases, you add the exponents.
For our given example:- We first see the exponents are \(4\) and \(3\).- To combine, simply perform the addition \(4 + 3 = 7\).
This addition merges these like terms, allowing us to express the product in a simplified scientific form: \[8.2 \times 10^{7}\].Understanding how exponents interact simplifies many complex calculations, making scientific notation both powerful and efficient for dealing with large-scale data.