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Chapter 10: Problem 33

Simplify. Write the answer with positive exponents only. $$6^{-1}$$

Short Answer

Expert verified
\( \frac{1}{6} \)

Step by step solution

01

Understand Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. The equation to transform a negative exponent to a positive one is: \[ a^{-n} = \frac{1}{a^n} \].
02

Rewrite the Expression

Apply the negative exponent rule: \[ 6^{-1} = \frac{1}{6^1} \].
03

Simplify the Expression

Writing it in its simplest form we get: \[ \frac{1}{6} \].

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Key Concepts

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

negative exponents
Negative exponents might seem tricky at first, but they're not too difficult to understand. Whenever you see a negative exponent, it means you need to take the reciprocal of the base and raise it to the corresponding positive exponent. For example, if you have the expression \( a^{-n} \), it can be transformed into \(\frac{1}{a^n}\).
This is because the negative sign tells us to invert the number, which is the reciprocal. Therefore, \( 6^{-1} \) becomes \(\frac{1}{6^1}\).
Remember, the negative exponent indicates that the base should be flipped to its reciprocal.
reciprocals
Reciprocals are important for understanding negative exponents. When you see a number or a variable with a negative exponent, you should immediately think of its reciprocal. The reciprocal of a number is simply \( \frac{1}{x} \), where \( x \) is the original number.
For instance, the reciprocal of 6 is \( \frac{1}{6} \). This concept is crucial for transforming expressions like \( 6^{-1} \) into positive exponent forms, which simplifies to \(\frac{1}{6}\). By knowing and applying reciprocals, you can turn any negative exponent into a positive one easily.
This helps simplify many mathematical expressions and makes calculations more straightforward.
simplifying expressions
Simplifying expressions involves removing any negative exponents and presenting the expression in its simplest form. Starting with an expression like \( 6^{-1} \), we use the rule of negative exponents to turn it into \(\frac{1}{6^1}\).
Simplifying further, the expression becomes \(\frac{1}{6}\). This is the simplest form of the expression with only positive exponents.
The key is to methodically apply the rules of exponents and reciprocals:
  • First, identify any negative exponents.
  • Next, apply the reciprocal to change them to positive exponents.
  • Then, simplify the resulting fraction if possible.
By following these steps, you can ensure any expression with negative exponents is fully simplified and easy to work with.

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