Question

Write the following numbers in standard notation. Use a terminal decimal point when needed. (a) \(6.500 \times 10^{3}\) (b) \(3.46 \times 10^{-5}\) (c) \(7.5 \times 10^{2}\) (d) \(1.8856 \times 10^{2}\).

Short answer

(a) 6500 (b) 0.0000346 (c) 750 (d) 188.56

Step-by-step solution

- Convert (a) to standard notation

To convert (a) \(6.500 \times 10^{3}\) to standard notation:i. Identify the exponent (3).ii. Move the decimal point of 6.500 three places to the right.This yields 6500.

- Convert (b) to standard notation

To convert (b) \(3.46 \times 10^{-5}\) to standard notation:i. Identify the exponent (-5).ii. Move the decimal point of 3.46 five places to the left.This yields 0.0000346.

- Convert (c) to standard notation

To convert (c) \(7.5 \times 10^{2}\) to standard notation:i. Identify the exponent (2).ii. Move the decimal point of 7.5 two places to the right.This yields 750.

- Convert (d) to standard notation

To convert (d) \(1.8856 \times 10^{2}\) to standard notation:i. Identify the exponent (2).ii. Move the decimal point of 1.8856 two places to the right.This yields 188.56.

Key concepts

decimal point movement

When converting numbers between scientific notation and standard notation, understanding how to move the decimal point is crucial. The decimal point's movement depends on the exponent of 10 in the scientific notation.

If the exponent is positive, move the decimal point to the right. For example, for the number \(6.500 \times 10^{3}\), the exponent 3 means you move the decimal point three places to the right, resulting in 6500.

If the exponent is negative, move the decimal point to the left. For instance, \(3.46 \times 10^{-5}\) has an exponent of -5, so you move the decimal point five places to the left, resulting in 0.0000346.

Remember:

• Positive exponent: move decimal point right.
• Negative exponent: move decimal point left.

standard notation

Standard notation is the regular way of writing numbers without using exponents. It's the format we use in everyday life. For example, instead of writing \(6.500 \times 10^{3}\), we simply write 6500.

When converting from scientific notation to standard notation, adjust the decimal point as necessary based on the exponent. If you had \(7.5 \times 10^{2}\), you move the decimal point two places to the right to get 750.

Standard notation is particularly useful for simplifying the reading and writing of large or small numbers. For instance, \(1.8856 \times 10^{2}\) becomes 188.56, making it easier to understand at a glance.

exponents in scientific notation

In scientific notation, exponents play a vital role in determining the decimal point’s position. The exponent tells you how many places and in which direction you need to move the decimal point.

An exponent of 3 in \(6.500 \times 10^{3}\) indicates moving the decimal point three places to the right, resulting in 6500.
An exponent of -5 in \(3.46 \times 10^{-5}\) indicates moving the decimal point five places to the left, resulting in 0.0000346.

The exponents simplify the representation of very large or very small numbers and make calculations more manageable. Always pay close attention to the sign of the exponent to correctly move the decimal point.

scientific notation to decimal

To convert from scientific notation to decimal form, follow these straightforward steps:

Identify the exponent of 10. Determine the direction to move the decimal point (right for positive exponents, left for negative exponents). Count the number of places to move the decimal point. Adjust the number accordingly.

Let's take a look at the provided examples. For \(7.5 \times 10^{2}\), move the decimal point two places to the right to get 750. For \(1.8856 \times 10^{2}\), move the decimal point two places to the right to get 188.56.

Understanding these conversions helps you work with both large and small numbers efficiently. With practice, it becomes more intuitive, aiding in various scientific and mathematical tasks.