Simplifying Expressions Using The Power Of A Power Rule

In the realm of mathematics, exponents play a crucial role in expressing repeated multiplication. When dealing with exponents, several rules and properties govern how we manipulate and simplify expressions. One such rule is the power of a power rule, which is the focus of our discussion today. This rule is essential for simplifying expressions where an exponential term is raised to another power. To thoroughly understand and correctly apply this rule, we will delve into its definition, explore examples, and analyze the given options to identify the one that accurately demonstrates the power of a power rule.

Grasping the Power of a Power Rule

The power of a power rule is a fundamental concept in algebra that simplifies expressions involving exponents raised to another power. This rule states that when you raise a power to another power, you multiply the exponents while keeping the base the same. Mathematically, it can be expressed as:

$(a^m)^n = a^{m imes n}$

where:

• a is the base,
• m is the inner exponent,
• n is the outer exponent.

This rule allows us to condense complex exponential expressions into simpler forms, making calculations and further manipulations easier. The essence of the rule lies in recognizing that raising a power to another power is equivalent to multiplying the exponents, which streamlines the simplification process. When faced with expressions like $(x^2)^3$, applying the power of a power rule directly yields $x^{2 imes 3} = x^6$, demonstrating its efficiency and elegance in simplifying exponential forms.

For example, let's consider the expression $(2^3)^2$. According to the power of a power rule, we multiply the exponents 3 and 2, keeping the base 2 the same. This gives us $2^{3 imes 2} = 2^6$. Calculating $2^6$, we get 64. This simple example illustrates how the power of a power rule works in practice, allowing us to simplify the expression efficiently. Understanding this rule is crucial for more complex algebraic manipulations and problem-solving involving exponents. The ability to quickly apply this rule not only simplifies expressions but also aids in grasping more advanced mathematical concepts that rely on exponent manipulation.

Analyzing the Given Options

To determine which option correctly applies the power of a power rule to simplify $(7^{10})^5$, let's analyze each option step by step:

• Option A: $(7^{10})^5 = 7^{10+5} = 7^2$

  This option incorrectly adds the exponents instead of multiplying them. The correct application of the power of a power rule requires multiplying the exponents, not adding them. Additionally, the final result of $7^2$ is incorrect given the erroneous initial step. This approach fundamentally misunderstands the rule, leading to an inaccurate simplification. Therefore, Option A does not demonstrate a proper understanding of the power of a power rule.

• Option B: $(7^{10})^5 = 7^{10-5} = 7^5$

  This option incorrectly subtracts the exponents. Subtraction of exponents is applicable in division, not when raising a power to another power. This is a clear misapplication of exponent rules. The correct procedure involves multiplying the exponents, making this option incorrect. Option B showcases a misunderstanding of the fundamental operation required when dealing with a power raised to a power, further highlighting the necessity of grasping the correct rule to avoid such errors.

• Option C: $(7^{10})^5 = 7^{10 imes 5} = 7^{50}$

  This option correctly multiplies the exponents, demonstrating a proper understanding of the power of a power rule. When a power is raised to another power, the exponents are indeed multiplied while the base remains the same. This option accurately simplifies the expression by performing the correct operation. Thus, Option C is the correct application of the power of a power rule. The result, $7^{50}$, is the accurate simplification of the given expression, showcasing a solid grasp of exponential rules.

• Option D: This option is a discussion category and does not provide an answer to the simplification problem.

By carefully examining each option, we can clearly see that Option C is the only one that correctly applies the power of a power rule. Options A and B demonstrate common errors in exponent manipulation, while Option D is simply a classification and not a solution.

Step-by-Step Application of the Power of a Power Rule

To further clarify the correct application of the power of a power rule, let's break down the simplification of $(7^{10})^5$ step by step:

1. Identify the base and exponents: In the expression $(7^{10})^5$, the base is 7, the inner exponent is 10, and the outer exponent is 5.
2. Apply the power of a power rule: According to the rule, $(a^m)^n = a^{m imes n}$. Therefore, $(7^{10})^5 = 7^{10 imes 5}$.
3. Multiply the exponents: Multiply the inner exponent (10) by the outer exponent (5) to get $10 imes 5 = 50$.
4. Write the simplified expression: Substitute the result back into the expression to get $7^{50}$.

Thus, the simplified form of $(7^{10})^5$ is $7^{50}$. This step-by-step process ensures a clear and accurate application of the power of a power rule, eliminating any confusion and solidifying the understanding of the concept. By following these steps, one can confidently simplify any expression involving a power raised to a power, further enhancing their proficiency in exponent manipulation.

Common Mistakes to Avoid

When applying the power of a power rule, it's essential to avoid common mistakes that can lead to incorrect simplifications. Here are some frequent errors to watch out for:

1. Adding Exponents Instead of Multiplying: A common mistake is adding the exponents when the power of a power rule requires multiplication. For example, incorrectly simplifying $(a^m)^n$ as $a^{m+n}$ instead of $a^{m imes n}$.
2. Subtracting Exponents: Another error is subtracting the exponents, which is applicable in division, not when raising a power to another power. This leads to incorrect simplifications and misunderstanding of the exponent rules.
3. Misunderstanding the Base: It's crucial to maintain the base throughout the simplification process. The power of a power rule only affects the exponents, not the base itself. Changing the base during simplification is a significant error.
4. Forgetting to Apply the Rule Completely: Sometimes, individuals may start the simplification process correctly but fail to complete it, leading to partially simplified expressions. Always ensure all applicable rules are applied to reach the simplest form.
5. Mixing Up Rules: Confusing the power of a power rule with other exponent rules, such as the product of powers rule or the quotient of powers rule, can lead to errors. Each rule has specific conditions under which it applies, and mixing them up results in incorrect simplifications.

By being aware of these common pitfalls and practicing the correct application of the power of a power rule, you can enhance your accuracy and confidence in simplifying exponential expressions. Avoiding these mistakes is key to mastering exponent manipulation and achieving correct results.

Conclusion

In conclusion, the correct application of the power of a power rule is essential for simplifying exponential expressions. The rule states that $(a^m)^n = a^{m imes n}$, meaning you multiply the exponents when raising a power to another power. Through our analysis, we have determined that Option C, which states $(7^{10})^5 = 7^{10 imes 5} = 7^{50}$, is the only option that correctly applies this rule. Options A and B demonstrate common errors, such as adding or subtracting exponents, while Option D is not a valid answer.

Understanding and accurately applying the power of a power rule is a fundamental skill in algebra. By following the correct steps and avoiding common mistakes, you can confidently simplify exponential expressions and excel in your mathematical endeavors. Mastering this rule not only aids in simplifying expressions but also provides a solid foundation for more advanced algebraic concepts. Thus, a thorough understanding of the power of a power rule is indispensable for success in mathematics.