Problem 61 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. There is a species of frogs in Cuba that attain a length of at most \(1.25 \times 10^{-2}\) meters.

Step-by-Step Solution

Answer

Question: Convert the length of a species of frog in Cuba given as \(1.25 \times 10^{-2}\) meters in scientific notation to standard decimal form. Answer: The length of the species of frog in Cuba, in standard decimal form, is 0.0125 meters.

Step 1: Understanding Scientific Notation

Scientific notation is a way to represent very small or very large numbers in a more compact and manageable manner. A number in scientific notation is written as the product of a decimal number between 1 and 10 (the mantissa) and a power of 10 (the exponent). In this problem, the number in scientific notation is \(1.25 \times 10^{-2}\), where 1.25 is the mantissa and -2 is the exponent.

Step 2: Calculate the Decimal Form

To convert this scientific notation into standard decimal form, we will multiply the mantissa (1.25) by 10 raised to the power of the exponent (-2). So, it will be calculated as follows: \(1.25 \times 10^{-2} = 1.25 \times (0.01) = 0.0125\)

Step 3: Write the Answer in Decimal Form

In standard decimal form, the length of the species of frog is 0.0125 meters.

Key Concepts

Converting Scientific Notation

Transforming a number from scientific notation to standard decimal form is a straightforward but crucial skill in mathematics and science. Scientific notation provides a concise way to represent numbers that are exceptionally large or small. To convert a number from scientific notation, like the example given with the Cuban frogs' length, \(1.25 \times 10^{-2}\), one multiplies the mantissa by 10 raised to the power of the exponent. Here's a step-by-step guide for conversion:

• Identify the mantissa, which is the number 1.25 in this case.
• Look at the exponent, -2, to determine the power of 10.
• Multiply the mantissa by 10 raised to the exponent.

This process shifts the decimal point of the mantissa two places to the left, because the exponent is negative, resulting in the standard decimal form 0.0125 meters for the frog length.

Standard Decimal Form

Standard decimal form is the way we usually write and see numbers, with no exponents involved. It shows the full value of the number on a scale that is easy to understand at a glance. When converting from scientific notation to standard decimal form, the aim is to simplify the expression so that it becomes easily readable. For example, \(1.25 \times 10^{-2}\) in scientific notation, once converted, becomes 0.0125. The key benefit of this form is its direct readability, allowing for straightforward comparison with other numbers without the need for calculation or translation of exponents.

Exponent

In the context of scientific notation, the exponent is a very efficient way to express the number of places the decimal must be moved. A positive exponent means the decimal moves to the right, which enlarges the number, while a negative exponent indicates a move to the left, making the number smaller. In our Cuban frogs' length example, the exponent is -2, which means we need to move the decimal point two places to the left. If the exponent were a positive 2 instead, we'd move the decimal two places to the right, making the mantissa 125 rather than 1.25.

Mantissa

The mantissa in scientific notation is the base number or the figure that is multiplied by 10 to the power of the exponent. When converted to standard decimal form, the mantissa dictates the precision of the number. It is always a value between 1 (inclusive) and 10 (exclusive). With the frogs' length, the mantissa is 1.25. It is important during conversion to keep in mind that the value of the mantissa is significant, as it carries the digits that convey the actual size or quantity before it’s scaled by the exponent.