Problem 66 - Algebra and Trigonometry

Question

Write each number in decimal notation. $$ 6.257 \times 10^{-10} $$

Step-by-Step Solution

Answer

0.0000000006257

Step 1: Understand the Scientific Notation

The given number is in scientific notation: \(6.257 \times 10^{-10}\). Scientific notation expresses numbers as a product of a coefficient (6.257 in this case) and a power of 10 (\(10^{-10}\)).

Step 2: Interpret the Power of 10

The exponent \(-10\) means you need to move the decimal point in the coefficient left by 10 places. Since the exponent is negative, it indicates that the number is very small, less than one.

Step 3: Write the Number in Decimal Form

Start with the number 6.257. Move the decimal point 10 places to the left. Each placeholder before the original number will be filled with zeros. 1. Original: 6.257 2. Move left 10 spaces: 0.0000000006257

Step 4: Final Check

Ensure that the number of zeros is correct by counting the digits moved left. Originally, there were 3 digits after the decimal in the number 6.257, so moving 10 places includes moving through those three and adding enough leading zeros to make up the rest.

Key Concepts

Exponent

In mathematics, an exponent is a small number placed to the upper right of a base number. It tells us how many times the base is to be multiplied by itself. For example, in the expression \(10^{-10}\), the number 10 is the base and \(-10\) is the exponent. Here, a negative exponent indicates a very small number, essentially functioning as a reciprocal multiplication. Instead of multiplying 10 ten times, we divide by 10 ten times. This effectively determines how far to move the decimal point in the number. Moving left for negative exponents and right for positive exponents.

An easy way to think about it is:

• Positive exponent: Increase the number (move decimal right)
• Negative exponent: Decrease the number (move decimal left)

By understanding the role of exponents in shifting the decimal, converting scientific notation into decimal form becomes intuitive.

Decimal Notation

Decimal notation is the standard way of writing numbers where each digit represents a power of ten. For example, the number 523 means \(5 \times 10^2 + 2 \times 10^1 + 3 \times 10^0\). It's a straightforward method for presenting numbers without exponents.

When converting from scientific notation to decimal notation, you move the decimal point towards the direction the exponent dictates. In our exercise, converting \(6.257 \times 10^{-10}\) to decimal involves moving the decimal point 10 positions to the left. This results in a very small number: 0.0000000006257.

Decimal notation is more readable in everyday applications compared to scientific notation, especially for very small or large numbers.

Power of 10

The power of 10 is an important part of scientific notation, representing how many times 10 is multiplied by itself. As seen in the expression \(10^{-10}\), exponents in the powers of 10 help determine the scale or size of a number.

• For example, \(10^3\) is 1,000 (ten cubed or ten times itself three times)
• \(10^{-2}\) means multiplying ten in reverse (1 divided by 10 twice), giving 0.01

The power of 10 essentially "scales" the base number, helping to easily express very large or very small numbers. Multiple powers of 10 reflect how much to shift the decimal point, crucial when converting between scientific and decimal notation. This concept makes scientific notation a powerful tool in both math and science, bringing clarity and efficiency in handling numbers across vast ranges.