Textbook Question
Simplify each expression. See Example 1. n⁶ • n⁴ • n
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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.3.23
Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. (2²)⁵
Verified step by step guidance1
Recognize that the expression is a power raised to another power: \((2^{2})^{5}\). According to the power of a power rule in exponents, when you raise a power to another power, you multiply the exponents.
Apply the power of a power rule: \((a^{m})^{n} = a^{m \times n}\). Here, \(a = 2\), \(m = 2\), and \(n = 5\).
Multiply the exponents: \(2 \times 5 = 10\).
Rewrite the expression using the new exponent: \$2^{10}$.
The expression is now simplified to \$2^{10}$. You can leave it in this exponential form unless you are asked to calculate the numerical value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponentiation and Power Rules
Exponentiation involves raising a base to a power, and power rules help simplify expressions with exponents. The key rule here is (a^m)^n = a^(m*n), which means when raising a power to another power, multiply the exponents.
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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem) Example 1
Properties of Real Numbers
Understanding that variables represent nonzero real numbers ensures that operations like exponentiation are valid and that no division by zero or undefined expressions occur. This context allows safe application of exponent rules.
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3:31Introduction to Complex Numbers
Simplification of Expressions
Simplification involves rewriting expressions in a more compact or standard form without changing their value. Applying exponent rules correctly reduces complex expressions like (2²)⁵ to a single power, making calculations easier.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
CONCEPT PREVIEW Work each problem. Match each polynomial in Column I with its factored form in Column II. I II a. 8x³ - 27 A. (3 - 2x) (9 + 6x + 4x²) b. 8x³ + 27 B. (2x - 3) (4x² + 6x + 9) c. 27 - 8x³ C. (2x + 3) (4x² - 6x + 9)
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
CONCEPT PREVIEW Which of the following is the correct factorization of x⁴ - 1? A. (x² - 1) (x² + 1) B. (x² + 1) (x + 1) (x - 1) C. (x² - 1)² D. (x - 1)² (x + 1)²
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Textbook Question
Simplify each expression. See Example 1. (½ mn) (8m²n²)
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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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