Problem 10 - College Algebra with Modeling and Visualization

Question

$$ 5 \times 10^{-3}=______ $$

Step-by-Step Solution

Answer

0.005

Step 1: Understanding the Problem

We need to express the product of the number 5 and the power of ten, \(10^{-3}\), not with multiplication, but as a standard decimal number.

Step 2: Calculating the Power of Ten

The expression \(10^{-3}\) can be rewritten in decimal form as \(0.001\)since \(10^{-3}\) means to move the decimal point three places to the leftfor the number \(1\).

Step 3: Performing Multiplication

Now we multiply 5 by \(0.001\):\[5 \times 0.001 = 0.005\]

Step 4: Writing the Final Answer

So, \(5 \times 10^{-3}\) equals \(0.005\) when written as a decimal.

Key Concepts

Decimal Conversion

Decimal conversion is a fundamental skill in mathematics, especially useful when dealing with scientific notation. To convert a scientific notation like \(10^{-3}\) to a decimal, you need to understand that negative exponents indicate a small number. Essentially, \(10^{-3}\) is equivalent to \(1 \div 10^3\). This requires shifting the decimal point of the number 1, three places to the left, transforming it into \(0.001\). When performing this conversion, keep these steps in mind:

• Count the exponent's value to determine how many places the decimal point must move. For \(10^{-3}\), move it three places to the left.
• If there are not enough numbers, fill in with zeros.

Understanding this conversion is crucial when dealing with scientific and mathematical problems as it allows you to seamlessly switch between notation forms.

Multiplication by Powers of Ten

Multiplication by powers of ten is a common operation performed when working with both small and large numbers in scientific notation. When multiplying a number by a power of ten, the number of places the decimal point moves depends on the power itself:

• For positive exponents, move the decimal to the right.
• For negative exponents, as in our example \(10^{-3}\), move the decimal to the left.

Let's consider multiplying 5 by \(10^{-3}\):

• First, convert \(10^{-3}\) to \(0.001\), as explained in the previous section.
• Then, multiply 5 by \(0.001\) to get \(0.005\).

This multiplication technique helps in estimating and calculating numbers quickly without the need for calculators, especially helpful in scientific and engineering fields.

Exponents

Exponents are a shorthand way to represent repeated multiplication of a base number. They are crucial for simplifying expressions and are especially helpful in expressing large or small numbers efficiently, like in scientific notation.In our problem, the exponent \(-3\) in \(10^{-3}\) tells us:

• The base is 10.
• The exponent \(-3\) indicates we need to perform repeated division by 10, three times.

This makes \(10^{-3}\) equivalent to \(1 \div 10 \times 10 \times 10\), ultimately leading to a very small decimal number. Exponents can be positive, like \(10^3 = 1000\), where the base number is multiplied by itself multiple times, or negative, focusing on division to attain fractional values. Understanding how exponents work facilitates the conversion between formats and simplifies complex calculations, leading to greater mathematical efficiency.