91Ó°ÊÓ

Exponents, also known as powers or indices, represent a way to express repeated multiplication of the same number. For example, when we write \(4^3\), it means that we are multiplying 4 by itself three times: \(4 \times 4 \times 4 = 64\).

Exponents have several important properties:

• Any number to the power of 0 is 1, so \(a^0 = 1\) for any non-zero number \(a\).
• Multiplying numbers with the same base adds the exponents: \(a^m \times a^n = a^{m+n}\).
• Dividing numbers with the same base subtracts the exponents: \(a^m / a^n = a^{m-n}\).
• Raising a power to another power multiplies the exponents: \((a^m)^n = a^{mn}\).

In our example, we were given the equation \(4^n = 65,536\). By understanding the properties and behavior of exponents, we could find that \(4^8 = 65,536\), showing that the exponent (or power) we needed is 8.

Positive integers are the set of all the whole numbers greater than zero: \(1, 2, 3, 4, \text{and so on}\). They are crucial in exponentiation because exponents are often applied to positive integers, especially in basic exercises.

In the context of our exercise, knowing that \(n\) is a positive integer means that it must be one of the counting numbers. Thus, our task was to find which positive integer \(n\) satisfies the equation \(4^n = 65,536\).

By examining the increasing powers of 4, we saw that when \(n = 8\), the equation is true: \(4^8 = 65,536\). Therefore, out of all positive integers, 8 is the only one that works for this equation.

An exponential equation is an equation where the unknown variable appears in the exponent. For instance, in our exercise, \(4^n = 65,536\), the variable \(n\) is in the exponent position.

Solving exponential equations often involves finding the value of the exponent that makes the equation true. This typically requires:

• Finding a common base on both sides of the equation or
• Using logarithms to handle more complex equations.

In simpler cases, such as our exercise, we can solve it by expressing 65,536 as a power of 4. We calculated the powers of 4 until we matched the value, discovering that \(4^8 = 65,536\). Therefore, the exponential equation is solved when \(n = 8\). This approach helps understand how exponential equations can be tackled, leading to mastery of solving for exponents.