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Decimal notation is a system of writing numbers that employs the use of a decimal point to separate the whole number part from the fractional part. It's the most common way of expressing numbers in everyday arithmetic and is the form we use for currency, measurements, and general counting.

In the context of scientific notation, converting a number like \(1 \times 10^{5}\) into decimal notation simply means rewriting it in a way that's more familiar and easier to understand. Here, 1 is the coefficient, and \(10^{5}\) represents the number 10 raised to the power of 5. To express \(1 \times 10^{5}\) in decimal notation, we interpret the exponent 5 as an instruction to move the decimal point in the coefficient 1 a total of five places to the right, filling in with zeroes as needed. Thus, \(1 \times 10^{5}\) becomes 100,000.

Exponents, also known as powers, are a shorthand way to express repeated multiplication of a number by itself. For example, \(10^{5}\) indicates that the number 10 should be multiplied by itself a total of four additional times (since the first instance of 10 is already there), leading to \(10 \times 10 \times 10 \times 10 \times 10\).

The number 10 is the base, and the number 5 is the exponent. In this case, since the exponent is positive, it directs us to move the decimal point to the right when converting to decimal notation. If the exponent were negative, it would mean the opposite: move the decimal point to the left. Exponents are a fundamental concept in mathematics, used widely in algebra, scientific calculations, and beyond.

Mathematical problem solving involves a step-by-step approach to understanding and solving mathematical expressions and equations. It's a critical skill that enables students to break down complex problems into manageable steps.

Take the example of converting \(1 \times 10^{5}\) from scientific to decimal notation. The process involves:

• Recognizing the expression is in scientific notation.
• Understanding the meaning of the exponent, which tells us how many places to move the decimal point.
• Performing the conversion by shifting the decimal point accordingly and filling in zeroes if necessary, leading to the final answer of 100,000.

This method of tackling problems is a cornerstone of all scientific and mathematical pursuits, helping students to organize their thoughts and reach solutions methodically.