As an expert in mathematics, I can provide a comprehensive explanation on how to calculate powers, including negative exponents. Calculating to the power is a fundamental operation in algebra and is used extensively in various fields of mathematics and science.

Step 1: Understanding Exponents

Exponents, also known as powers, are a way to represent repeated multiplication of a number by itself. The expression \( a^n \) means that 'a' is multiplied by itself 'n' times. For example, \( 2^3 \) means \( 2 \times 2 \times 2 \), which equals 8.

Step 2: Positive Exponents

When the exponent is a positive integer, the calculation is straightforward. You simply multiply the base number by itself the number of times indicated by the exponent. For instance:
- \( 3^2 \) means \( 3 \times 3 \), which equals 9.
- \( 4^3 \) means \( 4 \times 4 \times 4 \), which equals 64.

Step 3: Zero Exponent

Any non-zero number raised to the power of zero is 1. This is because any number multiplied by itself zero times has no effect, leaving you with the multiplicative identity, which is 1. For example:
- \( 5^0 = 1 \)
- \( (-2)^0 = 1 \)

Step 4: Negative Exponents

This is where the concept becomes a bit more nuanced. A negative exponent indicates that you should take the reciprocal of the base number raised to the corresponding positive exponent. In other words, \( a^{-n} = \frac{1}{a^n} \). For example:
- \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
- \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)

Step 5: Fractional Exponents

Fractional exponents involve roots. The expression \( a^{1/n} \) is equivalent to the nth root of 'a'. For example:
- \( 8^{1/3} \) is the cube root of 8, which is 2 because \( 2^3 = 8 \).
- \( 9^{1/2} \) is the square root of 9, which is 3 because \( 3^2 = 9 \).

Step 6: Complex Numbers and Exponents

Exponents can also be applied to complex numbers, which involve real and imaginary parts. The rules are a bit more involved and require understanding of Euler's formula and De Moivre's theorem.

Step 7: Scientific Notation

In scientific fields, large or small numbers are often expressed in scientific notation, which uses powers of 10. For example, \( 3.5 \times 10^3 \) means \( 3.5 \times 1000 \), or 3500.

Step 8: Practical Applications

Understanding how to calculate powers is crucial in various real-world applications, including calculating compound interest, determining the area and volume of geometric shapes, and solving exponential growth or decay problems.

Step 9: Technology and Software

In practice, many people use calculators or software to handle exponentiation, especially when dealing with large numbers or complex calculations. These tools can simplify the process and reduce the chance of error.

Step 10: Conclusion

Calculating to the power is a fundamental skill that builds upon the understanding of multiplication and the properties of numbers. It is used in a wide range of mathematical operations and has practical applications in everyday life. Whether you are a student learning algebra, a scientist conducting research, or a professional working with data, mastering the concept of exponentiation is essential.

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Calculate a negative exponent using inversion. When the exponent is a negative number, you are using the inverse of the number. For example, three to the power of negative four, or 3^-4, will be equal to one over three to the power of positive four (1/34) or 1/ 3 x 3 x 3 x 3 or 1/81.Apr 27, 2018read more >>