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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1:31 minutes

Problem 69

Textbook Question

Simplify using properties of exponents. [5x^(2/3)][4x^(1/4)]

Verified step by step guidance

Step 1: Identify the expression to simplify: \([5x^{2/3}][4x^{1/4}]\).

Step 2: Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\) to combine the exponents of \(x\).

Step 3: Add the exponents: \(\frac{2}{3} + \frac{1}{4}\).

Step 4: Find a common denominator to add the fractions: \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{1}{4} = \frac{3}{12}\).

Step 5: Combine the coefficients and the simplified exponent: \(5 \cdot 4 = 20\) and \(x^{\frac{8}{12} + \frac{3}{12}} = x^{\frac{11}{12}}\).

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Key Concepts

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

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these properties is essential for simplifying expressions with exponents.

Multiplying Monomials

Multiplying monomials involves combining coefficients and adding the exponents of like bases. For example, when multiplying 5x^(2/3) and 4x^(1/4), you multiply the coefficients (5 and 4) and apply the properties of exponents to the x terms. This process simplifies the expression into a single monomial.

Finding a Common Denominator

When dealing with exponents that have different fractional powers, finding a common denominator can facilitate simplification. This involves rewriting the exponents with a common base, allowing for easier addition or comparison. For instance, converting 2/3 and 1/4 to a common denominator helps in combining the exponents accurately.