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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 49
Chapter 1, Problem 49

Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 48/46

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Identify the expression to simplify: \(\frac{4^{8}}{4^{6}}\).
Recall the quotient rule for exponents, which states that for the same base \(a\), \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
Apply the quotient rule to the expression: \(\frac{4^{8}}{4^{6}} = 4^{8-6}\).
Simplify the exponent by subtracting: \(4^{8-6} = 4^{2}\).
Write the final expression without negative exponents (already done here): \$4^{2}$.

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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. For division, subtract the exponent in the denominator from the exponent in the numerator when the bases are the same, e.g., a^m / a^n = a^(m-n). This rule is essential for simplifying expressions like 4^8 / 4^6.
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Rational Exponents

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1 / a^n. Since the problem asks for answers without negative exponents, it is important to rewrite any negative exponents as positive by using this reciprocal rule.
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Zero and Negative Rules

Properties of Real Numbers

Understanding that variables represent nonzero real numbers ensures that division by zero does not occur and that exponent rules apply correctly. This assumption allows simplification without concern for undefined expressions or zero bases.
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Introduction to Complex Numbers
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