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In mathematics, exponents are used to express a number that is to be multiplied by itself a certain number of times. For example, the exponential notation \(2^3\) is shorthand for \(2 \times 2 \times 2\), which equals 8. The number 2 here is known as the 'base,' and the number 3 is the 'exponent.' When you encounter the expression \(10^2\), it means that the base 10 is multiplied by itself once, resulting in 100.

Exponents are not limited to just whole numbers or integers. They can represent large and small numbers in a compact form, which is particularly useful in scientific and engineering contexts where numbers can span many orders of magnitude. A crucial point when working with exponents is to remember that the base represents the repeated factor in the multiplication, and the exponent represents how many times that multiplication occurs.

Calculating exponents is a fundamental skill that involves repeated multiplication. To calculate an exponent, you multiply the base by itself as many times as indicated by the exponent. Consider the expression \(10^2\), where 10 is the base and 2 is the exponent. You would multiply 10 by itself once since the exponent is 2. The process is: \(10 \times 10 = 100\).

It's important to note that different rules apply when dealing with exponents that are negative, fractions, or zero. For instance, \(10^0\) equals 1, no matter what the base is (except when the base is also zero). Remembering these foundational rules will help you navigate through more complex calculations involving exponents without confusion.

Multiplying numbers in scientific notation combines two concepts: multiplication of the significant figures and addition of exponents. When you come across \(8.67 \times 10^2\), this is a number in scientific notation where \(8.67\) is the significant figure or 'coefficient,' and \(10^2\) represents that the coefficient should be considered 100 times larger.

The process of multiplication with exponents is streamlined by first multiplying the coefficients, and then simply multiplying the base 10 numbers which simply adds the exponents together in a case where the bases are the same, an operation that is simpler than it sounds due to the properties of exponents. For instance, \(10^2 \times 10^3\) would result in \(10^(2+3) = 10^5\), meaning you could just add the exponents if the bases are identical.