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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1:36 minutes

Problem 115

Textbook Question

Identify the property illustrated in each statement. Assume all variables represent real numbers. (t-6)*(1/t-6)=1, if t-6 not equal to 0

Verified step by step guidance

Identify the expression: \((t-6) \cdot \left(\frac{1}{t-6}\right) = 1\).

Recognize that the expression involves multiplication of a number and its reciprocal.

Recall the property: The multiplicative inverse property states that for any non-zero real number \(a\), \(a \cdot \frac{1}{a} = 1\).

Apply the property: Here, \(a = t-6\), and \(\frac{1}{a} = \frac{1}{t-6}\), so \((t-6) \cdot \left(\frac{1}{t-6}\right) = 1\) illustrates the multiplicative inverse property.

Note the condition: The property holds as long as \(t-6 \neq 0\), ensuring \(t-6\) is not zero, which would make the reciprocal undefined.

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Key Concepts

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

Properties of Real Numbers

Real numbers possess specific properties, such as the commutative, associative, and distributive properties. These properties govern how numbers interact in operations like addition and multiplication. Understanding these properties is essential for manipulating algebraic expressions and solving equations.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. This property is crucial when solving equations, as it allows us to set each factor to zero to find possible solutions. In the given statement, it helps in understanding the conditions under which the equation holds true.

Reciprocal Relationships

Reciprocal relationships involve pairs of numbers whose product is one. For any non-zero number 'a', its reciprocal is '1/a'. In the context of the given equation, recognizing that (t-6) and 1/(t-6) are reciprocals helps in simplifying the expression and understanding the conditions under which the equation is valid.