Problem 378 In the following exercises, simp... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 378

In the following exercises, simplify. $$\frac{x^{-6}}{x^{4}}$$

Short Answer

Expert verified

\frac{x^{-6}}{x^{4}} = \frac{1}{x^{10}}

Step by step solution

Apply the quotient rule for exponents

When dividing like bases, subtract the exponents. For the given expression $\frac{x^{-6}}{x^{4}}$, subtract the exponent in the denominator from the exponent in the numerator: $-6 - 4$

Simplify the exponent

Calculate the result of the subtraction: $-6 - 4 = -10$. This gives us $\frac{x^{-6}}{x^{4}} = x^{-10}$

Express the result as a positive exponent

Convert the negative exponent into a positive one by taking the reciprocal. Therefore, $x^{-10} = \frac{1}{x^{10}}$

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Key Concepts

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

Quotient Rule for Exponents

The quotient rule for exponents is a crucial concept when simplifying expressions involving exponents. This rule states that when you divide two expressions with the same base, you can subtract the exponent of the denominator from the exponent of the numerator. For example, in the exercise \[ \frac{x^{-6}}{x^{4}} \], the bases are the same (both are x). Using the quotient rule, we subtract the exponents: \[ -6 - 4 = -10 \]. Hence, \[ \frac{x^{-6}}{x^{4}} = x^{-10} \]. This result is an intermediate step that still needs further simplification.

Negative Exponents

Understanding negative exponents is essential for fully simplifying expressions. A negative exponent indicates that the base is on the wrong side of the fraction line. For instance, if you have \[ x^{-10} \], the negative exponent means you should take the reciprocal of the base and make the exponent positive. Hence, \[ x^{-10} = \frac{1}{x^{10}} \]. This conversion helps in simplifying and understanding the expression better. Remember, the negative sign in the exponent tells you to flip the base to the opposite side of the fraction.

Reciprocal of Exponents

The reciprocal of exponents is a concept directly related to negative exponents. When you see a negative exponent, it signals that you need to take the reciprocal of the base. For example, with \[ x^{-10} \], the negative exponent makes you flip the base from the numerator to the denominator, resulting in \[ \frac{1}{x^{10}} \]. In our exercise, after applying the quotient rule and simplifying to \[ x^{-10} \], the final step is to convert it to its reciprocal form: \[ \frac{1}{x^{10}} \]. This process ensures the exponent is positive and the expression is fully simplified.

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91影视

In the following exercises, simplify. $$\frac{x^{-6}}{x^{4}}$$

Short Answer

Step by step solution

Apply the quotient rule for exponents

Simplify the exponent

Express the result as a positive exponent

Key Concepts

Quotient Rule for Exponents

Negative Exponents

Reciprocal of Exponents

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