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Exponents are a powerful way to express repeated multiplication of a number by itself. In mathematical terms, an exponent indicates the number of times you multiply the base by itself. For example, in the expression \(3^4\), the base is 3 and the exponent is 4. This means that 3 is multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3\). This results in 81. Using exponents simplifies the expression and makes handling large numbers much easier.

When dealing with scientific notation, exponents play a crucial role. They help us efficiently express very large or very small numbers by using powers of ten. In the expression \(3.6 \times 10^4\), the exponent 4 indicates that the decimal point of the number 3.6 is moved 4 places to the right, making the number 36,000.

Multiplying powers is a fundamental rule in algebra that allows you to simplify expressions with exponents. When multiplying two powers with the same base, you add the exponents together. For instance, if you have \(a^m \times a^n\), the result is \(a^{m+n}\). This rule significantly simplifies computations involving powers.

In the original exercise, we see this rule applied with numbers in scientific notation. After squaring \(3 \times 10^8\) to get \(9 \times 10^{16}\), we multiplied the powers: \((3.6 \times 10^4) \times (9 \times 10^{16})\). Here, we added the exponents 4 and 16, according to the multiplication rule for powers, resulting in a new power of \(10^{20}\). By multiplying the coefficients \(3.6\) and \(9\), you determine the new overall coefficient for scientific notation.

Mathematical expressions are combinations of numbers, variables, and operations. They are the foundation for more complex mathematical analysis and problem-solving. Expressions can range from simple arithmetic combinations to intricate algebraic formulas.

In the exercise at hand, the expression \(mc^2\) involves both multiplication and exponentiation. The given values of \(m = 3.6 \times 10^4\) and \(c = 3 \times 10^8\) convert the general expression into a specific calculation. Evaluating mathematical expressions like these involves replacing variables with known values and applying mathematical rules like the order of operations.

Understanding the form and structure of mathematical expressions is key to evaluating them correctly. It involves identifying operations like squaring or multiplying variables and constants and then following through with the proper steps to simplify the entire expression.