Factor and simplify each algebraic expression. $16x^{-\frac{3}{4}}+32x^{\frac{1}{4}}$

Verified step by step guidance

1. Identify the common factors in the terms: look at the coefficients (16 and 32) and the variable parts with exponents (\(x^{-\frac{3}{4}}\) and \(x^{\frac{1}{4}}\)).
2. Factor out the greatest common factor (GCF) from the coefficients, which is 16, and also factor out the variable with the smallest exponent, which is \(x^{-\frac{3}{4}}\).
3. Rewrite the expression by factoring out the GCF: \(16x^{-\frac{3}{4}} \left( 1 + 2x^{\left( \frac{1}{4} - \left(-\frac{3}{4}\right) \right)} \right)\).
4. Simplify the exponent inside the parentheses by adding the exponents: \(\frac{1}{4} - \left(-\frac{3}{4}\right) = \frac{1}{4} + \frac{3}{4} = 1\).
5. Rewrite the expression inside the parentheses with the simplified exponent: \(16x^{-\frac{3}{4}} (1 + 2x)\), which is the factored and simplified form.

Key Concepts

Negative and Fractional Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, while fractional exponents represent roots. For example, x^(-3/4) means 1 divided by the fourth root of x cubed, and x^(1/4) means the fourth root of x.

Factoring Algebraic Expressions

Factoring involves rewriting an expression as a product of its factors. To factor expressions with terms involving exponents, identify the greatest common factor (GCF) including variables with the smallest exponents, then factor it out to simplify the expression.

Simplifying Expressions with Like Terms

Simplifying requires combining like terms, which have the same variable raised to the same power. When terms have different exponents, factoring out common factors helps rewrite the expression in a simpler form, even if terms cannot be directly combined.