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Scientific notation is a method of writing very large or very small numbers in a concise, standardized form. It is composed of two parts: a coefficient and a base of 10 raised to an exponent. The coefficient is a number greater than or equal to 1 and less than 10, while the base is always 10 in scientific notation.

For example, the scientific notation of 870 is written as \(8.7 \times 10^{2}\). Here, 8.7 acts as the coefficient and the exponent 2 tells us how many places to move the decimal point. If the exponent is positive, we move the decimal to the right, indicating a larger number. Conversely, a negative exponent means moving the decimal to the left, indicating a smaller number. Thus, scientific notation helps in simplifying the representation and calculation of both very large and minuscule numbers.

Exponents in mathematics, also known as powers, are a way of expressing repeated multiplication of a number by itself. The exponent is the small number located in the upper right corner of the base number and represents how many times the base number is multiplied by itself. For instance, \(10^{2}\) is equivalent to \(10 \times 10\), which equals 100.

Understanding exponents is fundamental when working with scientific notation. The exponent dictates the magnitude and direction (left or right) in which to shift the decimal point to convert scientific notation back to decimal form. A positive exponent indicates a number larger than one, while a negative exponent signifies a fractional value or a number smaller than one.

Place value refers to the position of a digit in a number that determines its value. In the decimal number system, each place represents a power of 10. For instance, in the number 870, the '8' is in the hundreds place, the '7' is in the tens place, and the '0' is in the ones place.

When converting from scientific notation to decimal notation, you use the place value concept to determine where to position the decimal point. For \(8.7 \times 10^{2}\), we adjust the decimal point two places to the right because of the exponent 2. Hence, the tens place value receives the 7, and the hundreds place value now holds the 8, producing the number 870.