Evaluating Exponents

An exponent represents the number of times a base number is multiplied by itself. It is denoted by a superscript number placed to the right and slightly above the base number. For example, in the expression 3⁴, 3 is the base and 4 is the exponent. This means that 3 is multiplied by itself 4 times.

Basic Exponent Rules

When evaluating exponents, there are some basic rules to keep in mind:

1. Product of Powers: When multiplying two numbers with the same base, you can add the exponents. For example, 3² * 3⁴ = 3⁶.
2. Quotient of Powers: When dividing two numbers with the same base, you can subtract the exponents. For example, 6⁵ / 6³ = 6².
3. Power of a Power: When a number is raised to an exponent and then raised to another exponent, you can multiply the exponents. For example, (2³)⁴ = 2¹².
4. Zero Exponent: Any non-zero number raised to the power of 0 is equal to 1. For example, 5⁰ = 1.
5. Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, 4^-2 = 1 / 4² = 1 / 16.

Examples of Evaluating Exponents

Let's go through some examples to demonstrate how to evaluate exponents using the rules mentioned above:

1. Evaluate 2³ * 2⁵:

Using the product of powers rule, we add the exponents: 2³ * 2⁵ = 2⁸.

2. Evaluate 9⁴ / 9²:

Using the quotient of powers rule, we subtract the exponents: 9⁴ / 9² = 9².

3. Evaluate (5²)³:

Using the power of a power rule, we multiply the exponents: (5²)³ = 5⁶.

4. Evaluate 7⁰:

As per the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1: 7⁰ = 1.

5. Evaluate 8^-2:

Using the negative exponent rule, we take the reciprocal of the base raised to the positive exponent: 8^-2 = 1 / 8² = 1 / 64.

Practice Problems

Now, let's try some practice problems to reinforce the concepts:

1. Evaluate 4³ * 4².
2. Evaluate 10⁵ / 10³.
3. Evaluate (6⁴)².
4. Evaluate 2⁰.
5. Evaluate 3^-3.

After solving these practice problems, you should have a better understanding of how to evaluate exponents using the rules and examples provided above.