Very large or very small numbers can be written in the following form, where a is an integer between 1 and 9 and b is an integer.

Question

a x 10^b

Estimate the numbers in the table. Then, find the values of a and b for each number.

Value in Standard Form Estimated Value
a × 10b
Estimated U.S. population in 2014 318,900,000 a =
, b =
Thickness of piece of metal in m 0.00053 a =
, b =

Answer 1

To express the given numbers in the form \( a \times 10^b \), we need to determine values for \( a \) and \( b \) for each case.

1. Estimated U.S. population in 2014: 318,900,000

To convert this to scientific notation:

• Identify \( a \): The leading digit between 1 and 9 is 3 (the first digit of 318,900,000).
• Identify \( b \): Count the number of places the decimal point must move to the left to place it after the 3.

From 318,900,000 (which can be thought of as 3.189 \(\times 10^8\)), we need to move the decimal point 8 places to the left to get it into the form \( 3.189 \times 10^8 \).

So, we can write:

• \( a = 3.189 \)
• \( b = 8 \)

2. Thickness of piece of metal in m: 0.00053

Convert this to scientific notation:

• Identify \( a \): The leading digit is 5 (from 0.00053). This number will also be written as \( 5.3 \times 10^{-4} \).
• Identify \( b \): The decimal point moves 4 places to the right to get from 0.00053 to 5.3.

Thus, we can write:

• \( a = 5.3 \)
• \( b = -4 \)

Summary:

1. Estimated U.S. population in 2014:

   • \( a = 3.189 \)
   • \( b = 8 \)

2. Thickness of piece of metal in m:

   • \( a = 5.3 \)
   • \( b = -4 \)