To raise a power of 10 to another power, you multiply the exponent by the power. Explain why this works.

Exponentiation is the mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. In other words, if we have a base number 'a' and an exponent 'n', exponentiation would give us the product of multiplying 'a' by itself 'n' times, which is denoted as \(a^n\). For instance, \(3^4\) means we multiply 3 by itself 4 times (3 x 3 x 3 x 3), resulting in 81.

This operation is fundamental in mathematics because it helps us understand complex phenomena such as exponential growth, which can be seen in population growth, compound interest, and more. When the base is a power of 10, exponentiation becomes particularly elegant, because the result is simply a 1 followed by 'n' number of zeros, where 'n' is the exponent. For example, \(10^3\) results in 1000.

Mathematical exponents are the 'n' in the expression \(a^n\), signifying how many times the base 'a' is used as a factor. One of the foundational properties of exponents is that when you have the same base raised to two powers multiplied together, you can add the exponents. For instance, \(a^m \times a^n = a^{m+n}\).

Zero and Negative Exponents

Special cases include zero and negative exponents. For any nonzero base 'a', the rule states that \(a^0 = 1\) and for negative exponents \(a^{-n} = \frac{1}{a^n}\). These properties allow exponents to express very large or very small numbers in a concise manner, making them invaluable in the fields of science and engineering.

Powers of ten are a key concept in scientific notation and a cornerstone of the decimal number system. Each power of ten represents a tenfold increase or decrease in quantity, depending on whether the exponent is positive or negative. For example, \(10^3\) equals 1000, and \(10^{-3}\) equals 0.001.

Exponentiation of powers of ten follows a pattern that is easy to grasp: a one followed by as many zeros as the exponent. It's a simple but powerful tool for representing large numbers like a trillion (\(10^{12}\)) or small numbers like a billionth (\(10^{-9}\)). This simplicity carries over when you have to perform operations like multiplication or division with powers of ten, as it usually involves shifting the decimal point to the right for multiplication or to the left for division by the number of the exponent.

When it comes to multiplying exponents with the same base, the process is straightforward: you simply add the exponents together. This rule, known as 'the Product of Powers' property, makes calculations more compact and manageable. For instance, if you have \(a^m \times a^n\), the result is \(a^{m+n}\).

Examples in Application

Multiplying exponents is particularly useful for simplifying expressions and solving equations. As an example, \((2^3)^4\) can be represented as \(2^{3\times4}\) or \(2^{12}\), which simplifies to 4096. Understanding this property aids in grasping more complex mathematical concepts and contributes to enhancing problem-solving skills across various levels of mathematics.