Textbook Question
Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. (r⁸/s²)³
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.3.13
Simplify each expression. See Example 1. n⁶ • n⁴ • n
Verified step by step guidance1
Identify the base and the exponents in the expression \(n^{6} \cdot n^{4} \cdot n\). Here, the base is \(n\) for all terms.
Recall the exponent rule for multiplication with the same base: \(a^{m} \cdot a^{n} = a^{m+n}\). This means you add the exponents when multiplying powers with the same base.
Rewrite the expression by explicitly showing the exponents: \(n^{6} \cdot n^{4} \cdot n^{1}\) (since \(n\) is the same as \(n^{1}\)).
Add the exponents together: \(6 + 4 + 1\) to get the new exponent.
Express the simplified form as \(n^{6+4+1}\), which is \(n^{11}\), but do not calculate the sum explicitly here as per instructions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Laws of Exponents
The laws of exponents govern how to simplify expressions involving powers of the same base. Specifically, when multiplying terms with the same base, you add their exponents. For example, n^a * n^b = n^(a+b). This rule is essential for simplifying expressions like n⁶ • n⁴ • n.
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Intro to Law of Cosines
Base Consistency in Exponentiation
When applying exponent rules, the base must be the same for the rules to hold. In the expression n⁶ • n⁴ • n, all terms share the base 'n', allowing the exponents to be combined. Recognizing consistent bases is crucial before performing operations on exponents.
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Simplification of Algebraic Expressions
Simplification involves rewriting expressions in a more compact or standard form without changing their value. For powers, this means combining like terms and reducing the expression to a single term when possible, such as combining n⁶ • n⁴ • n into n^(6+4+1) = n¹¹.
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Related Practice
Textbook Question
Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. (2²)⁵
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Textbook Question
CONCEPT PREVIEW Work each problem. Match each polynomial in Column I with its factored form in Column II. I II a. x² + 10xy + 25y² A. (x + 5y) (x - 5y) b. x² - 10xy + 25y² B. (x + 5y)² c. x² - 25y² C. (x - 5y)² d. 25y² - x² D. (5y + x) (5y - x)
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Textbook Question
Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. -(4m³n⁰)²
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Textbook Question
Simplify each expression. See Example 1. (-4x⁵) (4x²)
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Textbook Question
Simplify each expression. See Example 1. (-3m⁴) (6m²) (-4m⁵)