Problem 29 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 9^{0} \cdot 9^{2} $$

Step-by-Step Solution

Answer

Answer: 81

Step 1: Apply the Product Rule

Since we are multiplying two exponential expressions with the same base, we will apply the product rule by adding the exponents: $$ 9^{0} \cdot 9^{2} = 9^{0 + 2} $$

Step 2: Simplify the Exponents

Now, we will simplify the exponent 0 + 2 by performing the addition: $$ 9^{0 + 2} = 9^{2} $$

Step 3: Calculate the expression

Finally, calculate the value of the expression by raising the base to the power of the exponent: $$ 9^2 = 81 $$ So, $$ 9^{0} \cdot 9^{2} = 81 $$

Key Concepts

Product Rule of Exponents

When multiplying exponential expressions that have the same base, the product rule comes into play. This rule is quite straightforward: add the exponents while keeping the base unchanged. This is because multiplying powers of the same base is like multiplying the base by itself multiple times. Let's consider an example to clarify this concept.

Suppose you have the expression \( b^m \times b^n \), where \( b \) is the base and \( m \) and \( n \) are the exponents. According to the product rule, this is simplified as \( b^m \times b^n = b^{m+n} \).

• In our exercise, the base is 9 and it's raised to the power of 0 and 2, respectively.
• By applying the product rule, we add the exponents: \(9^{0} \times 9^{2} = 9^{0+2} \).
• This simplifies to \(9^2\).

Thus, understanding the product rule can significantly simplify the process of working with exponents in multiplication.

Quotient Rule of Exponents

Similar to the product rule, the quotient rule helps with the division of exponential expressions that have the same base. Instead of adding the exponents, the quotient rule tells us to subtract the exponent in the denominator from the exponent in the numerator. The formula is \( b^m ÷ b^n = b^{m-n} \), where \( b \) is the base and \( m \) and \( n \) are the exponents.

For instance, if you are given \( b^5 ÷ b^3 \), then according to the quotient rule, we subtract the exponents: \( b^5 ÷ b^3 = b^{5-3} = b^2 \).

• It is important to note that the base remains the same.
• The quotient rule is crucial for simplifying expressions when dividing like bases.

Exponential Expressions

An exponential expression is a mathematical notation involving two numbers, the base and the exponent. The expression \( b^n \) tells us to multiply the base \( b \) by itself \( n \) times. It's a concise way to represent repeated multiplication. These expressions become even more versatile when combined with the product and quotient rules of exponents, allowing complex calculations to be carried out with relative ease.

Take the base 9 in our example. The exponential expression \(9^2\) means 9 multiplied by itself once, since the exponent is 2. Exponential expressions can represent large or small numbers efficiently, especially in scientific notation where the base is 10. But remember, the exponent must be a whole number for these specific rules to apply.

Base and Exponent

The base and the exponent are the two key components in an exponential expression. The base is the value that is being multiplied by itself, and the exponent tells us how many times to multiply the base. In the expression \( b^n \), \( b \) represents the base and \( n \) is the exponent.

For example, if we look at the expression \(9^2\),

• 9 is the base, the number we are repeatedly multiplying,
• 2 is the exponent, indicating that we multiply 9 by itself once since the counting starts from the base as the first instance.

The beauty of bases and exponents lies in their simplicity and power, making it possible to work smoothly with very large or very small numbers and to simplify the expression of repeated multiplication.