Rational Exponents and Radical Functions
Lessons in this chapter
About Rational Exponents and Radical Functions
Rational Exponents and Radical Functions connects fractional exponents to roots, then builds up to graphing and solving with radicals.
n-th Roots and Rational Exponents introduces the connection between a fractional exponent and a root, before Properties of Rational Exponents and Radicals extends the exponent rules to work with these new rational exponents.
Graphing Radical Functions covers sketching and reading the graph of a function built from a root.
Solving Radical Equations and Inequalities closes the chapter with the techniques (and the extraneous-solution check they require) for solving equations and inequalities that contain a radical.
The four lessons, from n-th Roots and Rational Exponents through to Solving Radical Equations and Inequalities, each open as a worksheet worked in the page and scored as you answer. Properties of Rational Exponents and Radicals is the one worth repeating, since most later mistakes trace back to a misapplied exponent rule. Free to practice online.
Frequently asked questions
n-th Roots and Rational Exponents connects radicals to fractional exponents, so the same expression can be written either way. Properties of Rational Exponents and Radicals gives the rules for simplifying them, Graphing Radical Functions covers the shape, domain, and transformations of square root and cube root graphs, and Solving Radical Equations and Inequalities deals with isolating the radical and then checking for extraneous solutions. This sits in Number and Quantity (N) and Algebra (A) in the high school Common Core, with the graphing work in Functions (F).
A rational exponent like x^(1/n) means the n-th root of x: a fractional exponent and a root are two notations for exactly the same operation.
The same rules for integer exponents (multiplying same bases adds exponents, raising a power to a power multiplies them) still apply to rational exponents.
Identify the function’s domain first (since even roots of negative numbers aren’t real), then plot key points, paying attention to where the graph starts or is undefined.
Raise both sides of the equation to a power that eliminates the radical, solve normally, then check every solution in the original equation.
You must check for extraneous solutions, raising both sides to a power can introduce solutions that don't actually work in the original radical equation.