Problem 66 - Elementary Algebra

Question

Perform the following operations. $$ \left(2 \times 10^{-1}\right)\left(3 \times 10^{-5}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the following numbers in scientific notation: (2 x 10^(-1)) * (3 x 10^(-5)) Answer: 6 x 10^(-6)

Step 1: Multiply the decimal part of the numbers

First, we need to multiply the decimal parts of both numbers. So, we will multiply 2 by 3: $$ (2)(3) = 6 $$

Step 2: Add the exponents of powers of 10

In this step, we need to add the exponents of both powers of 10: $$ (10^{-1})(10^{-5}) = 10^{-1 + (-5)} = 10^{-1 - 5}. $$

Step 3: Simplify the result

Now, we put together what we obtained in Step 1 and Step 2: $$ \left(2 \times 10^{-1}\right)\left(3 \times 10^{-5}\right) = (6)(10^{-1 - 5})= 6 \times 10^{-6}. $$ So, the result of the multiplication is \(6 \times 10^{-6}\).

Key Concepts

Multiplying Powers of Ten

When multiplying powers of ten, it's crucial to understand how exponents work. Each power of ten is expressed as \(10^n\), where \(n\) is the exponent. This exponent tells us how many times to multiply 10 by itself.

For example, \(10^3 = 10 \times 10 \times 10 = 1000\).
When we multiply powers of ten, we simply add their exponents. For instance:

• \((10^a)(10^b) = 10^{a+b}\).

In the exercise, \(10^{-1}\) and \(10^{-5}\) means:

• \(10^{-1} = \frac{1}{10}\) and \(10^{-5} = \frac{1}{100000}\).

Adding these exponents results in \(10^{-6}\), which simplifies the multiplication.

Exponents

Exponents are a mathematical way to represent repeated multiplication. They consist of a base, like the number 10, and an exponent, which tells you how many times the base is used as a factor. For example, in \(10^4\), the base is 10 and the exponent is 4, so it means 10 multiplied by itself 4 times.

Exponents can also be negative, representing division.

• For example, \(10^{-3}\) is \(\frac{1}{10^3} = \frac{1}{1000}\).

When you see negative exponents, think of it as a way to indicate fractions. Understanding how exponents work helps simplify complex calculations, especially those involving scientific notation.

Multiplication of Decimal Numbers

Multiplying decimal numbers involves a couple of simple steps. First, ignore the decimal point and multiply the numbers as if they are whole numbers. For example, \(2\) and \(3\) are multiplied as \(2 \times 3 = 6\).

Once you have this product, determine how many decimal places were in the original numbers. If there was one in each, your product must have two decimal places. However, if multiplying within scientific notation, the placement of decimal points is automatically adjusted by the powers of ten.

In our exercise, the decimal multiplication was simple:

• \((2 \times 3) = 6\)

because it’s part of a scientific notation multiplication.