Question

Express these numbers in standard notation. a) \(1.381 \times 10^{5}\) b) \(5.22 \times 10^{-7}\) c) \(9.998 \times 10^{4}\)

Short answer

a) 138100 b) 0.000000522 c) 99980

Step-by-step solution

Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. It's written as the product of a number (usually between 1 and 10) and a power of ten. Here, we need to convert from scientific notation to standard notation.

Convert 1.381 × 10^5 to Standard Notation

For part a, the expression is given as \(1.381 \times 10^5\). The exponent 5 indicates that we need to move the decimal point 5 places to the right. This results in \(138100\).

Convert 5.22 × 10^{-7} to Standard Notation

For part b, the expression is given as \(5.22 \times 10^{-7}\). The negative exponent -7 indicates that we need to move the decimal point 7 places to the left. Filling in with zeros, this results in \(0.000000522\).

Convert 9.998 × 10^4 to Standard Notation

For part c, the expression is given as \(9.998 \times 10^4\). The exponent 4 indicates that we move the decimal point 4 places to the right, resulting in \(99980\).

Key concepts

Exponents

Exponents are a key component of scientific notation, playing a crucial role in simplifying numbers by representing how many times a number is multiplied by itself. For example, when you see an expression like \(10^5\), it means 10 multiplied by itself 5 times, which equals 100,000.
Exponents help us express both extremely large and small numbers efficiently without writing out countless zeroes.

• A positive exponent moves the decimal point to the right.
• A negative exponent shifts the decimal point to the left.

Understanding exponents is critical for tasks like converting numbers from scientific notation to standard notation. This allows for easier reading and comprehension of complex figures.

Standard Notation

Standard notation refers to the regular way of writing numbers without exponents. It is how we typically read and write numbers in everyday life. When converting from scientific to standard notation, you eliminate the power of ten and adjust the number accordingly.
In the example \(1.381 \times 10^{5}\),
the conversion process involves moving the decimal 5 places to the right, resulting in 138,100. This step-by-step adjustment is crucial:

• For positive exponents, extend the number to the right by moving the decimal right.
• Introduce zeros as necessary to fill the moved spaces.

Understanding standard notation is vital since it provides a simple way to express large or small numbers without special mathematical symbols.

Decimal Points

Decimal points are essential in accurately representing numbers, especially when shifting between scientific and standard notation. The placement of the decimal point determines a number's size and precision. In scientific notation, moving the decimal point by a specific number of places corresponds to the exponent value.
For instance, in \(5.22 \times 10^{-7}\),
the exponent of -7 means the decimal shifts 7 places to the left, resulting in 0.000000522. This involves inserting zeroes to ensure the position is correct.

• Positive exponents require moving the decimal point right and adding zeros after the original number.
• Negative exponents involve moving the decimal left and inserting zeros at the start.

Understanding the role of the decimal point ensures accuracy when converting numbers and avoids common mistakes such as incorrect number sizes.