Problem 51 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. $$ 67 \times 10^{3} $$

Step-by-Step Solution

Answer

The standard form of \(67 \times 10^3\) is \(6.7 \times 10^4\).

Step 1: Understand Standard Form

Standard form means expressing a number as a single-digit integer followed by decimals, if necessary, multiplied by a power of ten. Here, we are given a product of a whole number and a power of ten.

Step 2: Calculation of Product

Multiply the given number 67 by the factor of 10 raised to the third power. This entails multiplying 67 by 1000 (since \(10^3 = 1000\)).

Step 3: Perform the Multiplication

Calculate the result of \(67 \times 1000\). This results in 67000 as the value.

Step 4: Express in Standard Form

To write 67000 in standard form, reformat it as \(6.7 \times 10^4\). The number 67000 is expressed as a single-digit integer (6.7) times a power of ten.

Key Concepts

Understanding Powers of Ten

Powers of ten are a shorthand way to represent large or small numbers. The basic idea is multiplying ten by itself a certain number of times. The number of times you multiply ten is called the exponent. For example, in the term \(10^3\), the exponent is 3, which tells us to multiply 10 by itself three times, resulting in 1000.

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^5 = 100000\)

This notation efficiently expresses numbers like 1000 or 0.001 using exponents instead of writing out all the zeros. This simplification is especially useful in scientific and mathematical contexts, helping prevent errors by avoiding lengthy sequences of digits.

Single-Digit Integer's Role in Standard Form

In the context of standard form, a single-digit integer plays a crucial role. Standard form is a way of writing numbers that helps highlight the significance of each digit. Typically, a number is expressed as a product of a number between 1 and 10 and a power of ten.
For example, when expressing the number 67000 in standard form, we break it down to \(6.7 \times 10^4\). Here, 6.7 is our single or close-to-single digit integer, highlighting the most significant digit of the number.

• This form makes it easier to compare vastly different numbers
• Focuses on the most significant digits

By concentrating on a single-digit or close single-digit number, standard form enables swift and clear communication of data.

Mastering Multiplication in Standard Form

Multiplying numbers when expressing them in standard form involves a straightforward process. First, you perform the multiplication of the number by the power of ten. Let's look at how this works with our example \(67 \times 10^3\).

• Understand that \(10^3 = 1000\)
• Multiply 67 by 1000, which gives 67000

Next, convert this result from its expanded form into standard form. For 67000, it turns into \(6.7 \times 10^4\).

• Shift the decimal until the number is between 1 and 10 (6.7 here)
• Adjust the power of ten to reflect the number of decimal shifts (\(10^4\) now)

This step ensures the number is in a quickly readable format, preserving the value while accentuating its most influential figures, crucial for understanding and communicating in math and science.