Table of contents
- 0. Review of Algebra4h 16m
- 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 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Radical Expressions
4:36 minutes
Problem 1b
Textbook Question
Write ∛64 using exponents and evaluate.
Verified step by step guidance
1
Recognize that the cube root of a number can be expressed as an exponent: \( \sqrt[3]{64} = 64^{1/3} \).
Identify that 64 is a power of 2: \( 64 = 2^6 \).
Substitute \( 64 \) with \( 2^6 \) in the expression: \( (2^6)^{1/3} \).
Apply the power of a power property of exponents: \( (a^m)^n = a^{m \cdot n} \).
Calculate the new exponent: \( 2^{6 \cdot (1/3)} = 2^2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radicals and Exponents
Radicals are expressions that involve roots, such as square roots or cube roots. The cube root of a number 'a' is expressed as ∛a, which is equivalent to a raised to the power of 1/3. Understanding the relationship between radicals and exponents is crucial for manipulating and evaluating expressions involving roots.
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Evaluating Cube Roots
To evaluate a cube root, you need to find a number that, when multiplied by itself three times, equals the original number. For example, ∛64 asks for a number that satisfies x³ = 64. Recognizing that 4 × 4 × 4 = 64 allows us to conclude that ∛64 = 4.
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Properties of Exponents
Properties of exponents, such as the product of powers and power of a power, help simplify expressions involving exponents. When rewriting radicals as exponents, these properties can be applied to combine or simplify terms. For instance, knowing that a^(m/n) = ∛(a^m) can facilitate the evaluation of expressions like ∛64.
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