Domanda del libro di testo
In Exercises 103–114, factor completely. (x+y)4−100(x+y)2
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 1, Problema 113
In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (x−5/4y1/3x−3/4)−6
Guida verificata passo dopo passo1
Step 1: Start by simplifying the expression inside the parentheses. Combine the exponents of x using the property of exponents: a^m * a^n = a^(m+n). Here, x^(-5/4) * x^(-3/4) becomes x^(-5/4 - 3/4).
Step 2: Simplify the exponent of x. Add the fractions in the exponent: -5/4 - 3/4 = -8/4 = -2. So, the expression inside the parentheses becomes x^(-2) * y^(1/3).
Step 3: Apply the outer exponent of -6 to each term inside the parentheses using the property (a * b)^m = a^m * b^m. This gives (x^(-2))^(-6) * (y^(1/3))^(-6).
Step 4: Simplify each term by multiplying the exponents. For x^(-2)^(-6), multiply -2 and -6 to get x^(12). For y^(1/3)^(-6), multiply 1/3 and -6 to get y^(-2).
Step 5: Combine the simplified terms. The final expression is x^(12) * y^(-2). If needed, rewrite y^(-2) as 1/y^2 to express the result without negative exponents.
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Exponents and Negative Exponents
Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, x^(-a) = 1/(x^a). Understanding how to manipulate negative exponents is crucial for simplifying expressions involving them.
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Rational Exponents
Fractional Exponents
Fractional exponents indicate both a power and a root. For instance, x^(m/n) means the n-th root of x raised to the m-th power. This concept is essential for simplifying expressions that involve roots and powers simultaneously, allowing for a clearer understanding of the relationships between different terms.
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04:06Rational Exponents
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms, applying exponent rules, and reducing fractions. This process often requires careful attention to the order of operations and the properties of exponents. Mastery of simplification techniques is vital for solving complex algebraic problems efficiently.
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05:07Simplifying Algebraic Expressions
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