Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Exponents

College Algebra

0. Review of Algebra

Exponents

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1:22 minutes

Problem 51

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. x^12/x^8

Verified step by step guidance

Identify the expression to simplify: \( \frac{x^{12}}{x^8} \).

Apply the quotient rule for exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \).

Subtract the exponent in the denominator from the exponent in the numerator: \( 12 - 8 \).

Rewrite the expression using the result from the subtraction: \( x^{12-8} \).

Simplify the expression to \( x^4 \), ensuring there are no negative exponents.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponent Rules

Exponent rules are fundamental principles that govern the operations involving powers of numbers or variables. One key rule is that when dividing like bases, you subtract the exponents: a^m / a^n = a^(m-n). This rule is essential for simplifying expressions with exponents, as it allows for the reduction of terms efficiently.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/a^n. In the context of simplification, it is important to express answers without negative exponents, which means converting any negative exponent terms into their positive reciprocal forms.

Nonzero Variables

In algebra, it is often assumed that variables represent nonzero real numbers to avoid undefined expressions, such as division by zero. This assumption is crucial when simplifying expressions with exponents, as it ensures that all operations are valid and that the results remain within the realm of real numbers.