Problem 46 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. $$ -5.68 \times 10^{-1} $$

Step-by-Step Solution

Answer

-0.568

Step 1: Understand the problem

The question asks us to write a number that is currently in scientific notation into its standard form. The number given is \(-5.68 \times 10^{-1}\).

Step 2: Know the scientific notation properties

Scientific notation is written as \(a \times 10^n\), where \(a\) is a decimal number, and \(n\) determines the direction and number of decimal places shifted. Here, \(n = -1\), indicating a shift of one decimal place to the left.

Step 3: Convert to standard form

To convert the number from scientific notation to standard form, move the decimal point in \(-5.68\) one place to the left as indicated by the exponent \(-1\). This changes the number to \(-0.568\).

Key Concepts

Standard Form

Standard form is a way of expressing numbers that makes them easy to read and understand at a glance. It's important for presenting data in a clear and concise manner. In standard form, numbers are written in their usual numeric format without the use of exponents. This is particularly useful for understanding the actual size or magnitude of the number. For example, if you encounter a number like \(-5.68 \times 10^{-1}\) in scientific notation, you can convert it to standard form for immediate comprehension, which results in \(-0.568\). This allows for a straightforward interpretation of the value and facilitates quick comparisons between numbers.

Decimal Shifting

Decimal shifting is the process that defines how you move the decimal point in a number, governed by the exponent in scientific notation. In scientific notation like \(a \times 10^n\), the exponent \(n\) tells you how many places to move the decimal point.

• If \(n\) is positive, move the decimal to the right.
• If \(n\) is negative, move it to the left.

For the number \(-5.68 \times 10^{-1}\), the decimal is moved one place to the left, thus converting it to \(-0.568\) in standard form.Understanding decimal shifting is essential, especially when handling very small or large numbers, as it keeps the representation compact yet efficient.

Exponents

Exponents are numbers written as superscripts that indicate how many times a number, known as the base, is multiplied by itself. In the context of scientific notation, the base is typically \(10\). Thus, exponents help in swiftly representing numbers that are extremely large or small.A positive exponent means the decimal point moves to the right, creating larger numbers, while a negative exponent means the decimal point moves to the left, creating smaller numbers.In the given problem, \(-5.68 \times 10^{-1}\), the exponent \(-1\) indicates a single move of the decimal to the left. This leads to the conversion of the number in standard form to \(-0.568\). Exponents are powerful tools in mathematics that enhance efficiency and clarity, especially when dealing with calculations over multiple magnitudes.