Question

Convert these to nonscientific notation: (a) \(1.52 \times\) \(10^{4},\) (b) \(7.78 \times 10^{-8}\)

Short answer

(a) \(1.52 \times 10^{4}\) in nonscientific notation is 15200, and (b) \(7.78 \times 10^{-8}\) in nonscientific notation is 0.0000000778.

Step-by-step solution

Conversion of (a)

The number \(1.52 \times 10^{4}\) has a positive exponent, indicating that it is greater than 1. Convert this by moving the decimal point to the right according to the value of the exponent, which is 4 in this case. That results in \(15,200\).

Conversion of (b)

The number \(7.78 \times 10^{-8}\) has a negative exponent, indicating that it is a fraction, less than 1. Convert this by moving the decimal point to the left according to the value of the exponent, which is 8 in this case. This results in \(0.0000000778\). Remember to pad with zeros accordingly.

Key concepts

Positive Exponents

When the exponent in scientific notation is positive, it signifies that the number is greater than 1. This is a key concept to grasp. The positive exponent tells us how many places to move the decimal point to the right.
For instance, in the number \(1.52 \times 10^{4}\), the exponent is 4. This means you move the decimal point 4 places to the right. It’s like multiplying by 10, four times in a row. So 1.52 becomes 15200.

• Start with the original number: 1.52
• Since the exponent is 4, move the decimal 4 places right
• Fill in any gaps with zeros as needed to get 15,200

Positive exponents are straightforward—think of them as multiplying, which makes a smaller number larger.

Negative Exponents

Negative exponents are the opposite of positive ones. They indicate that the number is a fraction, less than 1. With negative exponents, the decimal point moves to the left.
Consider the example \(7.78 \times 10^{-8}\). The exponent \(-8\) means you shift the decimal point left by 8 places. This involves adding leading zeros to help move the decimal.

• Start with the original number: 7.78
• Move the decimal 8 places to the left, as the exponent is \(-8\)
• Insert the necessary zeros to form 0.0000000778

Understanding negative exponents is crucial—it's like dividing or making the number smaller. Each move left makes the number one-tenth as big as before.

Decimal Point Conversion

Decimal point conversion is the process of shifting the decimal point to represent the number in standard decimal form from scientific notation. This skill is essential for interpreting scientific data in everyday numbers.
To convert a number in scientific notation to decimal form, assess the number of spaces to move the decimal:

• Positive exponent: Move the decimal to the right
• Negative exponent: Move it to the left
• In both cases, ensure to adjust by adding zeros where gaps occur

Utilizing these rules, scientific notation becomes less intimidating and much more manageable, allowing you to translate scientific numbers into forms you deal with every day.