In Exercises 91–100, simplify using properties of exponents. (20x^1/2)/(5x^1/4)

Verified step by step guidance

1. Identify the expression: $\frac{20x^{1/2}}{5x^{1/4}}$.
2. Simplify the coefficients: $\frac{20}{5}=4$.
3. Apply the quotient rule for exponents: $x^a/x^b=x^{a-b}$.
4. Subtract the exponents: $x^{1/2-1/4}$.
5. Combine the simplified coefficient and the result of the exponent subtraction: $4x^{1/4}$.

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these properties is essential for simplifying expressions with exponents.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form by dividing the numerator and denominator by their greatest common factor. In the context of exponents, this means applying the properties of exponents to both the coefficients and the variable parts of the fraction, allowing for a clearer and more manageable expression.

Radicals and Rational Exponents

Radicals and rational exponents are closely related concepts in algebra. A square root can be expressed as a rational exponent, such that √x = x^(1/2). This relationship allows for the manipulation of expressions involving roots using the same exponent rules, facilitating the simplification of complex expressions that include both radicals and exponents.