In Exercises 93–102, factor and simplify each algebraic expression. x^3/4−x^1/4

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Identify the common factor in the terms of the expression. In this case, the common factor is the smallest power of x, which is x^(1/4).

Factor out x^(1/4) from the expression. This means rewriting the expression as x^(1/4) * (x^(3/4 - 1/4) - 1).

Simplify the exponent inside the parentheses. Subtract the exponents in the term x^(3/4 - 1/4), which simplifies to x^(2/4) or x^(1/2).

Rewrite the factored expression as x^(1/4) * (x^(1/2) - 1).

Verify that the factored form is correct by distributing x^(1/4) back into the parentheses to ensure it matches the original expression.

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Factoring Algebraic Expressions

Factoring involves rewriting an expression as a product of its factors. In the context of polynomials, this often means identifying common terms or using special formulas, such as the difference of squares or perfect square trinomials. Understanding how to factor is essential for simplifying expressions and solving equations.

Exponents and Radicals

Exponents represent repeated multiplication of a base, while radicals are the inverse operation, indicating roots. In the expression x^(3/4) - x^(1/4), the fractional exponents can be interpreted as roots, which can help in factoring. Mastery of exponent rules is crucial for manipulating and simplifying expressions involving powers.

Common Factor Extraction

Common factor extraction is a technique used to simplify expressions by identifying and factoring out the greatest common factor (GCF) from all terms. In the given expression, both terms share a common factor of x^(1/4), which can be factored out to simplify the expression significantly. This concept is foundational in algebra for simplifying complex expressions.

In Exercises 93–102, factor and simplify each algebraic expression. x3/4−x1/4