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Simplify by reducing the index of the radical.
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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 108
Factor and simplify each algebraic expression.
Guida verificata passo dopo passo1
Start by rewriting the expression clearly: \(12x^{-\frac{1}{2}} + 6x^{-\frac{3}{2}}\).
Identify the common factor in both terms. Look at the coefficients (12 and 6) and the powers of \(x\) (\(x^{-\frac{1}{2}}\) and \(x^{-\frac{3}{2}}\)).
Factor out the greatest common factor (GCF). The GCF of 12 and 6 is 6, and for the powers of \(x\), take the smaller exponent \(x^{-\frac{3}{2}}\) as the common factor.
Rewrite each term inside the parentheses by dividing the original terms by the GCF: \(6x^{-\frac{3}{2}} \left( \frac{12x^{-\frac{1}{2}}}{6x^{-\frac{3}{2}}} + \frac{6x^{-\frac{3}{2}}}{6x^{-\frac{3}{2}}} \right)\).
Simplify the terms inside the parentheses by subtracting exponents when dividing powers of \(x\) and simplifying coefficients, resulting in a factored and simplified expression.
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Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, x^(-n) equals 1 divided by x^n. Understanding this helps in rewriting and simplifying expressions involving negative powers.
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Zero and Negative Rules
Factoring Algebraic Expressions
Factoring involves finding the greatest common factor (GCF) of terms and expressing the expression as a product of factors. This process simplifies expressions and makes further operations easier, such as combining like terms or simplifying.
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05:09Introduction to Algebraic Expressions
Combining Like Terms with Exponents
When terms have the same base and exponent, they can be combined by adding or subtracting their coefficients. Recognizing like terms with fractional or negative exponents is essential for simplifying algebraic expressions correctly.
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7:39Introduction to Exponent Rules
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