Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, convert each number in scientific notation to standard notation. \(9.8 \times 10^{-9}\)

Step-by-Step Solution

Answer

The standard notation is 0.0000000098.

Step 1: Understand Scientific Notation

Scientific notation is a way to express numbers as a product of a number between 1 and 10 and a power of 10. In this notation, the number is written as \(a \times 10^n\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer.

Step 2: Identify the Components

The given number is \(9.8 \times 10^{-9}\). Here, \(9.8\) is the number between 1 and 10, and \(-9\) is the exponent. The exponent \(-9\) tells us that the decimal point in 9.8 needs to be moved 9 places to the left.

Step 3: Move the Decimal Point

Start with the number 9.8. Move the decimal point 9 places to the left. Since \(-9\) is negative, this means making the number smaller. Add zeros as necessary to fill in the gaps.

Step 4: Write the Number in Standard Notation

Moving the decimal point 9 places to the left in 9.8 gives \(0.0000000098\). This is the number in standard notation.

Key Concepts

Standard Notation

Understanding standard notation is crucial for translating numbers from scientific notation. Standard notation is the way we normally write numbers down, using digits and not exponents. It is a straightforward representation without

• any components expressed as powers of ten,
• without logarithmic expressions,
• and typically includes all the zeros instead of grouping them into a shorthand.

For instance, when converting from scientific to standard notation, you simply display the number with its full series of digits, including any necessary leading or trailing zeros. It's particularly useful in addressing very large or very small quantities in a more comprehensible numeric form.
In our example, the scientific notation of the number, which is given as \(9.8 \times 10^{-9}\), is converted to a standard one by ensuring every decimal is accurately represented, resulting in \(0.0000000098\). This long form helps visualize the actual size of the number.

Exponents

Exponents are a key concept when dealing with numbers in scientific notation. An exponent tells us how many times to multiply a number by 10. It's that superscript number that dictates the movement of the decimal point.When an exponent is positive, it signifies that the number is large, and the decimal point moves to the right. Conversely, a negative exponent, like

• \(-9\) for our example \(9.8 \times 10^{-9}\),
• indicates a small number,
• pushing the decimal point to the left.

Utilizing exponents simplifies complex numbers, making them compact and easier to manage. Understanding this power of 10 concept allows you to interpret scaling factors and size efficiently.
Remember that an exponent directly influences how the decimal adjusts, shaping the transition from scientific to standard notation.

Decimal Point Movement

Decimal point movement is a practical step in converting between scientific and standard notation. Once armed with knowledge about exponents, you can easily determine how to shift your decimal point.For instance, with \(9.8 \times 10^{-9}\), the exponent \(-9\) directs you to move the decimal point nine places to the left. This movement indicates making the number smaller by

• placing it in the context of standard notation,
• showing its true scale without using exponentiation.

The process involves adding zeros as placeholders to fill the empty spaces left as the decimal slides across.
Initially, you start at 9.8, move left, and after nine places, you find that the number in standard notation is \(0.0000000098\). This simple yet effective movement unravels the mathematical shorthand into a clear representation for better understanding.