Problem 118 - Elementary Algebra

Question

Problem: Evaluate \(\left(9 \times 10^{-5}\right)\left(3 \times 10^{-7}\right)\). Write the answer in scientific notation. $$ \begin{aligned} \text { Incorrect Answer: } &\left(9 \times 10^{-5}\right)\left(3 \times 10^{-7}\right) \\ &=27 \times 10^{-12} \\ &=2.7 \times 10^{-13} \end{aligned} $$

Step-by-Step Solution

Answer

\(2.7 \times 10^{-11}\)

Step 1: Multiply the coefficients

First, multiply the coefficients of the two numbers. Here, the coefficients are 9 and 3.\[9 \times 3 = 27\]

Step 2: Add the exponents

Next, add the exponents of the powers of 10. The exponents are -5 and -7.\[10^{-5} \times 10^{-7} = 10^{-5 + (-7)} = 10^{-12}\]

Step 3: Combine the results

Combine the results from steps 1 and 2.\[27 \times 10^{-12}\]

Step 4: Convert to scientific notation

Finally, convert the result to proper scientific notation. In scientific notation, the coefficient must be between 1 and 10. So,\[27 \times 10^{-12} = 2.7 \times 10^{-11}\]

Key Concepts

Multiplying Powers of 10

When we multiply numbers in scientific notation, we need to handle the coefficients (the numbers before the powers of 10) and the powers of 10 separately. Let’s break it down step by step.

• Step 1: Multiply the coefficients. In the original problem, the coefficients are 9 and 3. Multiply these two values:

\[9 \times 3 = 27\]

• Step 2: Add the exponents of the powers of 10.

The exponents in the problem are -5 and -7. So, you will add these two values together:\[10^{-5} \times 10^{-7} = 10^{-5 + (-7)} = 10^{-12}\]
Remember, when multiplying powers of 10, simply add the exponents together.

Adding Exponents

Adding exponents is straightforward, but it's critical to understand for working with scientific notation. Here’s the rule:

• When you multiply two powers of 10, you add their exponents.

In our problem, we had to deal with \[10^{-5} \times 10^{-7}\]So, you add the exponents:

• \(-5 + (-7) = -12\)

This gives you:\[10^{-12}\]
Just make sure to keep the base (10) the same and only add the exponents. This concept will help you simplify calculations quickly when dealing with powers of 10.

Scientific Notation

Scientific notation is a method for writing very large or very small numbers conveniently. The format for scientific notation is:
\[a \times 10^b\] where 1 ≤ \|a\| < 10 and b is an integer.

• Step 1: To write a number in scientific notation, place the decimal after the first significant digit and count the number of places the decimal has moved. The coefficient (a) must be between 1 and 10.
• Step 2: The exponent (b) tells how many places the decimal moved. If you move the decimal to the left, b is positive. If you move it to the right, b is negative.

Let’s apply scientific notation to our problem:

• We had: 27 \times 10^{-12}

We need to adjust 27 to be between 1 and 10. Adjusting 27 gives us 2.7 (as 27 = 2.7 \times 10). To balance this change, we add 1 to the exponent:\[27 \times 10^{-12} = (2.7 \times 10^1) \times 10^{-12}\]\[= 2.7 \times 10^{1-12} = 2.7 \times 10^{-11}\]Our final result in proper scientific notation is:\[2.7 \times 10^{-11}\]