Problem 86 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(9 \times 10^{-5}\right)\left(2 \times 10^{-1}\right) $$

Step-by-Step Solution

Answer

Question: Calculate the product of two numbers in scientific notation: $(9 \times 10^{-5})(2 \times 10^{-1})$. Answer: The product of the two numbers is $1.8 \times 10^{-5}$.

Step 1: Multiply the coefficients

First, we need to multiply the coefficients of the two numbers: $$ 9 \times 2 = 18 $$

Step 2: Add the exponents of base 10

Next, we sum the exponents of the two numbers: $$ -5 + (-1) = -6 $$

Step 3: Write the result in scientific notation

Now, we have the coefficient 18 and the exponent -6. To write the result in scientific notation, we combine the coefficient and the exponent, like this: $$ 18 \times 10^{-6} $$ However, since a number in scientific notation should have a coefficient between 1 and 10, we need to adjust the coefficient and exponent by one, like this: $$ 1.8 \times 10^{-5} $$ So, the final result is: $$ \left(9 \times 10^{-5}\right)\left(2 \times 10^{-1}\right) = 1.8 \times 10^{-5} $$

Key Concepts

Multiplying Coefficients

When dealing with scientific notation, you often encounter a situation where you need to multiply two numbers. An essential step is to multiply their coefficients. Coefficients in scientific notation simply refer to the real number part before the multiplication by a power of ten. For example, in the expression \(9 \times 10^{-5}\), the coefficient is 9.
To multiply coefficients, you treat them just like you would with regular numbers.

• Take the coefficients of each number involved in the multiplication.
• Multiply them together: \(9 \times 2 = 18\).

The result here is straightforward since multiplying coefficients doesn't change even if they are part of a larger scientific expression.

Adding Exponents

The next step is handling the exponents, which are the powers of ten in the scientific notation. When multiplying numbers in scientific notation, you add these exponents.
Consider the following exponents from the example: \(-5\) and \(-1\).

• The base of 10 remains the same, and we focus on the exponents.
• Add the exponents together: \(-5 + (-1) = -6\).

Adding exponents is a direct process, similar to basic addition with real numbers; just be careful to track positive and negative signs. This step adjusts the power of 10 in the result.

Adjusting Scientific Notation

Once you have multiplied the coefficients and summed the exponents, it’s time to check if the result fits the standard form of scientific notation: a coefficient between 1 and 10 followed by a power of ten.
In the example, we ended up with \(18 \times 10^{-6}\). Since 18 is not between 1 and 10, we need to make an adjustment:

• Divide the coefficient by 10 to get 1.8, making it fit within the desired range.
• Accordingly, increase the exponent by 1: from \(-6\) to \(-5\), to compensate for dividing 18 by 10.

Now, the result \(1.8 \times 10^{-5}\) is in standard scientific notation. Adjusting the scientific notation ensures the final number follows a consistent format, crucial for clear mathematical communication.