Understanding exponent rules is crucial to solving this problem. The expression r9/r3 involves the division of two terms with the same base (r). A fundamental rule of exponents states that when dividing exponential expressions with the same base, you subtract the exponents. This rule can be for...
Applying the Exponent Rule
Applying this rule to our expression r9/r3, we subtract the exponent in the denominator (3) from the exponent in the numerator (9): 9 - 3 = 6. Therefore, the simplified expression becomes r6.
Understanding the Base
It's important to remember that the base 'r' remains unchanged throughout the simplification process. The exponent rule only affects the exponents themselves, not the base. The base represents a variable or a constant value, and its value remains consistent during the operation.
Examples with Numerical Bases
Let's consider some examples with numerical bases to illustrate the concept. If we had 25/22, applying the rule would give us 25-2 = 23 = 8. Similarly, 104/101 simplifies to 104-1 = 103 = 1000. These examples reinforce the principle of subtracting exponents when dividing terms with the same base.
Avoiding Common Mistakes
A common mistake is to incorrectly add or multiply the exponents when dividing exponential expressions. Remember, the rule specifically states that we *subtract* the exponents when dividing terms with the same base. Adding or multiplying the exponents will lead to an incorrect result. Another potential pitfall is forgetting to maintain the base throughout the calculation; the base remains consistent throughout the simplification process.
Expanding the Expression (Alternative Approach)
We can also understand this simplification by expanding the expression. r9 represents r multiplied by itself nine times (r * r * r * r * r * r * r * r * r), and r3 represents r multiplied by itself three times (r * r * r). When we divide r9 by r3, we are essentially canceling out three 'r's from both the numerator and the denominator, leaving us with r multiplied by itself six times, which is r6.
r9/r3 = (r * r * r * r * r * r * r * r * r) / (r * r * r) = r6
Applications in Algebra and Calculus
Understanding and applying this exponent rule is fundamental in various areas of mathematics. In algebra, simplifying expressions involving exponents is a crucial skill. In calculus, this rule is used extensively in differentiation and integration, particularly when working with power functions. Mastering this concept lays a strong foundation for further mathematical studies.
Further Exploration: Negative Exponents
The concept extends further when dealing with negative exponents. For instance, if we had r3/r9, applying the rule would yield r3-9 = r-6. Negative exponents represent reciprocals; therefore, r-6 is equivalent to 1/r6. This demonstrates how the same fundamental rule applies even when dealing with negative exponents.
Practice Problems
To solidify your understanding, try solving similar problems. For example, simplify the following expressions: x7/x2, y12/y5, z4/z4 (this case results in z0 which equals 1), and a5/a8 (resulting in a negative exponent). These exercises will help you reinforce the concepts discussed.
Conclusion
In conclusion, the expression r9/r3 is equivalent to r6. This simplification is achieved by applying the fundamental rule of exponents for division: subtract the exponents when dividing exponential expressions with the same base. Understanding this rule and its applications is essential for mastering algebraic and calculus concepts.