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Chapter 1: Problem 64

Write an equivalent expression with negative exponents. $$ (-6)^{4} $$

Short Answer

Expert verified
\(\frac{1}{(-6)^{-4}}\).

Step by step solution

01

Understand Negative Exponents

Negative exponents represent the reciprocal of the positive exponent. For example, \((a^{-n}) = \frac{1}{a^n}\). To rewrite a positive exponent with a negative exponent, you can use this information.
02

Rewrite the Expression with Negative Exponents

Given \((-6)^4\), you can express this with a negative exponent by taking the reciprocal. The equivalent expression with a negative exponent is: \(\frac{1}{(-6)^-4}\).
03

Verify the Expression

To confirm the equivalence, recognize that \(\frac{1}{(-6)^-4}\) indeed simplifies back to \((-6)^4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
When dealing with exponents, it鈥檚 essential to understand the concept of a reciprocal. The reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). When we talk about negative exponents, we are actually using the concept of reciprocals.
The rule is: \(a^{-n} = \frac{1}{a^n}\). This means that any number raised to a negative exponent is equal to one divided by that number raised to the opposite positive exponent.
  • Example: \((2^{-3} = \frac{1}{2^3} = \frac{1}{8})\)

In our exercise, we need to express \((-6)^4\) using negative exponents. By the reciprocal rule, we get \(\frac{1}{(-6)^{-4}}\).
Equivalent Expressions
An equivalent expression is a way of writing the same value or quantity in a different form. In mathematics, equivalent expressions are crucial for simplifying problems and finding solutions.
When we rewrite \((-6)^4\) as \(\frac{1}{(-6)^{-4}}\), we are creating an equivalent expression. Even though they look different, both expressions represent the same value. To verify that these two expressions are equivalent, you can reverse the operation.
  • If \frac{1}{(-6)^{-4}}\ simplifies back to \((-6)^4\), then they are indeed equivalent.

Remember that the goal of using equivalent expressions is not to change the value but to provide a form that might be more useful in a given context.
Positive Exponents
When working with exponents, the starting point is usually positive exponents. These indicate how many times to multiply the base number by itself. For example, \(3^2\) means multiplying 3 by itself: \(3 \times 3 = 9\).
Positive exponents are easier to understand because they reflect straightforward multiplication. In our problem, \((-6)^4\) means multiplying -6 by itself four times, which results in 1296.
  • Step-by-step: \(-6 \times -6 = 36\), \(36 \times -6 = -216\), \(-216 \times -6 = 1296\).
By converting it to a negative exponent, we don't change the value but instead use a different form. This helps us understand the linkage between positive and negative exponents.
Recognizing these patterns is vital for solving more complex mathematical problems.

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