Problem 24 - Elementary Algebra

Question

\(\left(8 \times 10^{-4}\right)\left(2 \times 10^{-5}\right)\)

Step-by-Step Solution

Answer

16 \times 10^{-9}

Step 1 - Write the numbers in standard form

Start by identifying the two numbers in standard form: First number: \[8 \times 10^{-4} \]Second number: \[2 \times 10^{-5} \]

Step 2 - Multiply the coefficients

Multiply the coefficients (the numbers in front): \(8\) and \(2\).\[8 \times 2 = 16 \]

Step 3 - Add the exponents

Add the exponents of the powers of 10: \(-4\) and \(-5\).\[-4 + (-5) = -9 \]

Step 4 - Combine the results

Combine the product of the coefficients and the sum of the exponents to form the final answer: \[16 \times 10^{-9} \]

Key Concepts

Multiplying Coefficients

When dealing with scientific notation multiplication, the first step is to multiply the coefficients. Coefficients are the numbers that come in front of the powers of 10. In our exercise, the coefficients are 8 and 2.
These coefficients are straightforward to multiply because they are regular numbers, not affected by the powers of 10. Simply multiply them using basic arithmetic:

• Firstly, take the coefficient 8.
• Then, take the coefficient 2.
• Multiply them together: \[ 8 \times 2 = 16 \]

This result forms the coefficient part of your final answer. Remember, always handle the coefficients first before moving on to the exponents.

Adding Exponents

Once you have multiplied the coefficients, the next step is to add the exponents of each power of 10. Exponents tell you how many times to multiply 10 by itself. In this case, we have the exponents -4 and -5.
Here’s how to combine them:

• Take the exponent from the first number, which is -4.
• Then, take the exponent from the second number, which is -5.
• Add these exponents together: \[ -4 + (-5) = -9 \]

The result, -9, is now the exponent for the power of 10 in your final answer. Adding exponents is as simple as adding ordinary numbers, but it's important to keep track of the signs.

Standard Form

Finally, combine both parts you've worked out—the new coefficient and the new exponent—to express the answer in standard form. Standard form for scientific notation looks like this: \[ a \times 10^b \] where \(a\) is the coefficient, and \(b\) is the exponent.
In our example:

• The coefficient we found is 16.
• The exponent we found is -9.
• Putting them together gives us: \[ 16 \times 10^{-9} \]

This final answer maintains the proper format for scientific notation. By always following the steps—multiplying coefficients, adding exponents, and then combining them—you ensure accuracy and clarity in your solutions.