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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 67
Chapter 1, Problem 67

Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. (3pq)q2/6p2q4

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Start by writing the expression clearly: \(\frac{(3pq)q^{2}}{6p^{2}q^{4}}\).
Distribute the exponent on the numerator: \((3pq)q^{2} = 3p q \cdot q^{2} = 3p q^{1+2} = 3p q^{3}\).
Rewrite the expression with simplified numerator: \(\frac{3p q^{3}}{6 p^{2} q^{4}}\).
Separate the fraction into coefficients and variables: \(\frac{3}{6} \times \frac{p}{p^{2}} \times \frac{q^{3}}{q^{4}}\).
Simplify each part using exponent rules: \(\frac{3}{6} = \frac{1}{2}\), \(\frac{p}{p^{2}} = p^{1-2} = p^{-1}\), and \(\frac{q^{3}}{q^{4}} = q^{3-4} = q^{-1}\). Then rewrite without negative exponents by moving terms with negative exponents to the denominator.

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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 multiply, divide, and simplify expressions with powers. Key rules include adding exponents when multiplying like bases, subtracting exponents when dividing, and understanding zero and negative exponents. These rules help simplify expressions like (3pq)q² / (6p²q⁴).
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Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the numerator and denominator by canceling common factors. This process often requires factoring expressions and applying exponent rules to combine like terms, resulting in a simpler expression without negative exponents.
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Writing Expressions Without Negative Exponents

Negative exponents indicate reciprocals, so expressions with negative exponents must be rewritten by moving factors between numerator and denominator. For example, x⁻² becomes 1/x². This ensures the final answer is expressed with only positive exponents, as required.
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