Equivalent Expressions For 7^{-6} * 7^0

This article aims to provide a comprehensive explanation of how to find an expression equivalent to 7^{-6} ⋅ 7^0. We will delve into the fundamental exponent rules and apply them step-by-step to simplify the given expression. Furthermore, we will evaluate each of the provided options (A, B, C, and D) to determine which one matches our simplified result. This guide is designed for students and anyone looking to enhance their understanding of exponent manipulation and simplification in mathematics. Let's embark on this mathematical journey together to master the art of simplifying expressions with negative and zero exponents.

Simplifying the Initial Expression

To begin, let's simplify the given expression, 7^{-6} ⋅ 7^0. We need to recall two crucial exponent rules. The first rule is that any non-zero number raised to the power of 0 is equal to 1. In mathematical terms, this is expressed as a^0 = 1 for any a ≠ 0. Applying this rule, we find that 7^0 = 1. The second rule we need is the product of powers rule, which states that when multiplying powers with the same base, we add the exponents. This rule is represented as a^m ⋅ a^n = a^{m+n}. However, in this specific case, since 7^0 = 1, the expression simplifies considerably.

Substituting 7^0 with 1 in the original expression, we get 7^{-6} ⋅ 1. Any number multiplied by 1 remains the same, so the expression simplifies further to 7^{-6}. Now, we need to remember the rule for negative exponents, which states that a^{-n} = 1/a^n. Applying this rule to 7^{-6}, we rewrite it as 1/7^6. This simplified form will be our benchmark as we evaluate the given options to determine which one is equivalent to the original expression. Understanding these fundamental exponent rules is crucial for simplifying various algebraic expressions effectively. Next, we will analyze each of the provided options to identify the one that matches our simplified expression.

Evaluating the Options

Now that we have simplified the original expression to 1/7^6, let's evaluate each of the provided options to determine which one is equivalent. This involves applying exponent rules and simplifying each option to its simplest form. We will meticulously break down each step to ensure clarity and understanding.

Option A: (7⁷⁾{-7}

Option A is given as (7⁷⁾-7}**. To simplify this, we need to apply the power of a power rule, which states that **(a^m)n = a^{m⋅n}. In this case, we have a power (7^7) raised to another power (-7_**). Applying the rule, we multiply the exponents 7 * -7 = -49. Therefore, the expression becomes **_7^{-49. Using the negative exponent rule (a^{-n} = 1/a^n), we can rewrite this as 1/7^{49}. Comparing this to our simplified original expression 1/7^6, we can clearly see that they are not the same. Thus, option A is not equivalent to the original expression.

Option B: 7/7^7

Option B is given as 7/7^7. Here, we can consider the numerator, 7, as 7^1. So, the expression can be rewritten as 7^1/77. To simplify this, we use the quotient of powers rule, which states that a^m/an = a^m-n}_**. Applying this rule, we subtract the exponents 1 - 7 = -6. Therefore, the expression simplifies to **_7^{-6. Now, we apply the negative exponent rule (a^{-n} = 1/a^n) to rewrite 7^{-6} as 1/7^6. Comparing this to our simplified original expression, 1/7^6, we find that they are indeed the same. Hence, option B is equivalent to the original expression.

Option C: 7^0/7{-1}

Option C is given as 7^0/7{-1}. We already know that any non-zero number raised to the power of 0 is 1, so 7^0 = 1. The expression now becomes 1/7^{-1}. To simplify further, we recall the negative exponent rule, which states that a^{-n} = 1/a^n. Thus, 7^{-1} is equal to 1/7. Substituting this back into the expression, we get 1/(1/7). Dividing by a fraction is the same as multiplying by its reciprocal, so 1/(1/7) = 1 * 7 = 7. We can express 7 as 7^1. Comparing this to our simplified original expression, 1/7^6, it is evident that they are not equivalent. Therefore, option C is not the correct answer.

Option D: 1/7^{-6}

Option D is given as 1/7^{-6}. To simplify this, we again use the negative exponent rule, a^{-n} = 1/a^n. In this case, we have 7^{-6} in the denominator. We know that 7^{-6} is equal to 1/7^6. So, the expression becomes 1/(1/7^6). As we discussed earlier, dividing by a fraction is the same as multiplying by its reciprocal. Therefore, 1/(1/7^6) = 1 * 7^6 = 7^6. Comparing this to our simplified original expression, 1/7^6, it is clear that they are not the same. Thus, option D is not equivalent to the original expression.

Conclusion: The Equivalent Expression

After meticulously evaluating all the options, we have determined that option B, 7/7^7, is the expression equivalent to 7^{-6} ⋅ 7^0. We systematically simplified the original expression to 1/7^6 and then analyzed each option using the fundamental rules of exponents. Option B simplified to 7^{-6}, which is the same as 1/7^6, confirming its equivalence.

This exercise underscores the importance of understanding and applying exponent rules correctly. The rules for multiplying powers with the same base, dividing powers with the same base, the power of a power, and negative exponents are crucial tools in simplifying algebraic expressions. By mastering these rules, students can confidently tackle similar problems and deepen their understanding of mathematical concepts. The ability to manipulate exponents is not only essential for algebra but also forms a foundation for more advanced topics in mathematics and science. Therefore, consistent practice and a clear grasp of these rules are invaluable for academic success and beyond.