Problem 37 - 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. $$ k^{8} k^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(k^8 \cdot k^3\). Answer: \(k^{11}\)

Step 1: Recall the product rule of exponents

The product rule of exponents states that when you multiply two exponential expressions with the same base, you can add the exponents. Mathematically, for any base a and exponents m and n: $$ a^m \cdot a^n = a^{m + n} $$

Step 2: Apply the product rule to the given expression

We have the expression: $$ k^8 \cdot k^3 $$ Since the bases (k) are the same, we can apply the product rule of exponents and add the exponents (8 and 3) together: $$ k^8 \cdot k^3 = k^{8 + 3} $$

Step 3: Solve for the final expression

Now, we simply need to add the exponents (8 and 3) to find the final expression: $$ k^{8 + 3} = k^{11} $$ The simplified expression is \(k^{11}\).

Key Concepts

Understanding the Product Rule

When dealing with exponents, the product rule is a handy tool to simplify expressions. This rule states that if you multiply two expressions with the same base, you simply add their exponents together. For example, consider the expression \(a^m \times a^n\). Here, the base \(a\) remains constant, and the product rule tells us to add the exponents \(m\) and \(n\) together, resulting in \(a^{m+n}\).
This rule is particularly useful when working with larger exponents, as it simplifies calculations and reduces potential for error. It's important to note that the product rule only applies when the bases are identical. If the bases were different, this rule would not work.

• Example: \(k^8 \times k^3 = k^{8+3} = k^{11}\)
• Remember: Only add exponents if the bases match exactly.

Exploring the Quotient Rule

The quotient rule is another fundamental concept in the realm of exponents. It is crucial for simplifying expressions where one exponential term is divided by another with the same base. Understanding this rule makes it easier to work with expressions where division is involved.
The quotient rule states that for any base \(b\), if you have \(b^m \div b^n\), you subtract the exponent in the denominator from the exponent in the numerator, resulting in \(b^{m-n}\). This operation effectively simplifies the expression to a single exponent.
It's important to be cautious with subtraction, as negative exponents may occur if the exponent in the denominator is larger. Such results indicate reciprocals or inverse powers. Proper application of the quotient rule can drastically simplify complex fractions.

• Example: \(b^7 \div b^5 = b^{7-5} = b^2\)
• Use subtraction, but watch for negative exponents.

Simplification of Expressions with Exponents

Simplification of expressions with exponents often involves applying multiple rules such as the product and quotient rules. The goal is to reduce expressions to their simplest form, making them easier to understand and work with.
When simplifying, always start by addressing the operations indicated in the expression, whether it’s multiplication, division, or both. The key is to maintain the order of operations, tackling the same bases first and applying the appropriate rule.
If an expression involves large exponents, careful application of the rules helps to avoid mistakes. It's beneficial to rewrite expressions step by step, ensuring that each transformation is based on these fundamental rules:

• Combine like terms using the product rule.
• Simplify divisions using the quotient rule.
• Always check your work by ensuring that the expression cannot be simplified further.

Throughout the simplification process, remember that all bases need to be non-zero in order to apply these rules appropriately. By consistently practicing these steps, complex formulas turn into easier-to-handle equivalents, enhancing overall mathematical proficiency.