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Chapter 6: Problem 548

Simplify. $$\frac{x^{-7}}{x^{-3}}$$

Short Answer

Expert verified
\frac{1}{x^4}

Step by step solution

01

Understand the problem

The problem is to simplify the expression \(\frac{x^{-7}}{x^{-3}}\). This involves combining exponents using the rules of exponents.
02

Apply the quotient rule for exponents

The quotient rule states \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to the given expression: \(\frac{x^{-7}}{x^{-3}} = x^{-7 - (-3)} = x^{-7 + 3}\).
03

Simplify the exponent

Combine the exponents: \(-7 + 3\) to get \(-4\). Thus the expression simplifies to \(x^{-4}\).
04

Express the result with positive exponents

Recall that \(x^{-n} = \frac{1}{x^n}\). Apply this to the simplified expression: \(x^{-4} = \frac{1}{x^4}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form. This often includes combining like terms, using arithmetic operations, and applying algebraic rules.

In our exercise, we started with the expression \(\frac{x^{-7}}{x^{-3}}\). To simplify it, we used the quotient rule for exponents. By doing this, we made the expression easier to understand and work with. Simplification not only makes algebraic operations more manageable but also helps in solving equations efficiently.
Negative Exponents
Negative exponents can be tricky, but they follow specific rules that make them easier to handle once understood. A negative exponent indicates that the base is on the wrong side of the fraction line. For example, \(x^{-n} = \frac{1}{x^n}\).

In our example, we simplified \(x^{-4}\) to \(\frac{1}{x^4}\). This is because a negative exponent tells us to take the reciprocal of the base raised to the corresponding positive exponent. Recognizing and converting negative exponents is crucial for accurate simplification and solving of algebraic expressions.
Algebra Rules
Algebra rules are the backbone of simplifying expressions and solving equations. They encompass a variety of principles, including:

- **Quotient Rule**: To divide like bases with exponents, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
- **Negative Exponent Rule**: To convert a negative exponent into a positive one, take the reciprocal of the base: \(a^{-n} = \frac{1}{a^n}\).

In the exercise, these rules were used to simplify the expression \(\frac{x^{-7}}{x^{-3}}\) to \(\frac{1}{x^4}\).

By understanding and applying these rules, algebra becomes much more straightforward. With practice, using these rules will become second nature.

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Most popular questions from this chapter

Determine the degree of each polynomial. (a) \(14-29 x\) (b) \(z^{2}-5 z-6\) (c) \(y^{3}-8 y^{2}+2 y-16\) (d)\(23 a b^{2}-14\) (e) -3

In \(2010,\) the number of Facebook users each day who changed their status to 'engaged' was \(2 \times 10^{4}\). Convert this number to decimal form.

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