Quotient of Powers Examples

When dealing with powers of the same base, the quotient of powers rule states that we can simplify by subtracting the exponents. This rule applies to both integers and variables, making it a fundamental concept in algebra and mathematics. To understand this concept better, let’s delve into several examples that demonstrate how to apply the quotient of powers rule in different scenarios.

Example 1: Simplifying with Integers

Consider the expression ( \frac{4^6}{4^3} ). Here, both the numerator and the denominator have the same base, which is 4. According to the quotient of powers rule, when we divide two powers with the same base, we subtract the exponents. Thus, we have:

[ \frac{4^6}{4^3} = 4^{6-3} = 4^3 ]

This simplifies to ( 64 ), because ( 4^3 = 4 \times 4 \times 4 = 64 ).

Example 2: Applying with Variables

The quotient of powers rule also applies to variables. For instance, given the expression ( \frac{x^9}{x^4} ), applying the rule, we subtract the exponents:

[ \frac{x^9}{x^4} = x^{9-4} = x^5 ]

Example 3: Simplifying Expressions with the Same Base

Look at the expression ( \frac{3^8}{3^2} ). Since both parts of the fraction are powers of the same base (3), we can simplify by subtracting the exponents:

[ \frac{3^8}{3^2} = 3^{8-2} = 3^6 ]

To find the numerical value, we calculate ( 3^6 ), which equals ( 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729 ).

Example 4: Real-World Application

Consider a real-world scenario where the quotient of powers rule can be applied. Suppose a company’s profit grows at a rate of ( 2^5 ) (32 times) in the first year and then at a rate of ( 2^3 ) (8 times) in the second year, compared to the initial investment. To find the overall growth factor from the first to the second year relative to the first year’s growth, we can calculate the quotient:

[ \frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4 ]

This means the growth in the second year is 4 times the growth in the first year relative to the initial investment.

Example 5: Negative Exponents

The quotient of powers rule also applies when dealing with negative exponents. For instance, consider ( \frac{5^{-3}}{5^{-6}} ). Applying the rule:

[ \frac{5^{-3}}{5^{-6}} = 5^{-3 - (-6)} = 5^{-3 + 6} = 5^3 ]

This simplifies to ( 125 ), because ( 5^3 = 5 \times 5 \times 5 = 125 ).

Example 6: Variable Expressions with Coefficients

Sometimes, expressions may include coefficients. For example, ( \frac{2x^4}{x^2} ) simplifies by first applying the quotient of powers rule to the variables:

[ \frac{2x^4}{x^2} = 2x^{4-2} = 2x^2 ]

The coefficient 2 remains unchanged while the variable part simplifies according to the rule.

Conclusion

In conclusion, the quotient of powers rule is a powerful tool for simplifying expressions in mathematics, allowing us to easily handle and manipulate powers of the same base. By understanding and applying this rule, we can simplify complex expressions, solve equations, and model real-world phenomena with greater ease and accuracy. Whether dealing with integers, variables, or a combination of both, the quotient of powers rule provides a straightforward method for simplification, as demonstrated through the various examples provided.