Radical Functions and Equations

You have seen square roots before. You know that $\sqrt{16} = 4$ because $4 \times 4 = 16$. But what about $\sqrt[3]{8}$? Or $\sqrt[4]{81}$? And what happens when you put these roots into a function, like $f(x) = \sqrt{x - 3}$? How do you graph it? What values of $x$ are even allowed?

These are the questions we tackle in this lesson. And here is a comforting thought: if you understand square roots, you already have the foundation for everything that follows. Cube roots, fourth roots, and beyond all work the same way. Even better, there is a secret connection between radicals and exponents that makes everything cleaner once you see it.

Core Concepts

What Is a Radical Function?

A radical function is simply a function that involves a root. The most common is the square root function:

$$f(x) = \sqrt{x}$$

But we can have cube roots, fourth roots, or any root we like:

$$f(x) = \sqrt[3]{x} \quad \text{(cube root function)}$$ $$f(x) = \sqrt[4]{x} \quad \text{(fourth root function)}$$ $$f(x) = \sqrt[n]{x} \quad \text{(}n\text{th root function)}$$

The small number tucked into the radical symbol is called the index. For square roots, the index is 2, but we usually do not write it. When you see $\sqrt{x}$ with no index showing, it means $\sqrt[2]{x}$.

What does the $n$th root actually ask? It asks: “What number, when raised to the $n$th power, gives me this?” In other words:

$$\sqrt[n]{a} = b \quad \text{means} \quad b^n = a$$

So $\sqrt[3]{8} = 2$ because $2^3 = 8$. And $\sqrt[4]{81} = 3$ because $3^4 = 81$.

Domain of Radical Functions: Even vs. Odd Roots

Here is where things get interesting. The domain of a radical function (read: which $x$ values you are allowed to use) depends entirely on whether the index is even or odd.

Even roots (square roots, fourth roots, sixth roots, etc.) require the radicand to be non-negative. Why? Because there is no real number that you can square (or raise to any even power) to get a negative result. Try it: what number squared gives you $-4$? There is no real answer.

So for $f(x) = \sqrt{x}$, the domain is $x \geq 0$.

For $f(x) = \sqrt{2x - 6}$, you need $2x - 6 \geq 0$, which means $x \geq 3$.

Odd roots (cube roots, fifth roots, seventh roots, etc.) can handle any real number, including negatives. This is because negative numbers raised to odd powers stay negative. For instance, $(-2)^3 = -8$, so $\sqrt[3]{-8} = -2$. That works just fine.

So for $f(x) = \sqrt[3]{x}$, the domain is all real numbers.

This distinction is crucial when you start graphing these functions.

Graphing Radical Functions

The basic shapes of radical functions depend on whether the index is even or odd.

The basic square root function $f(x) = \sqrt{x}$:

• Starts at the origin (0, 0)
• Only exists for $x \geq 0$ (moves to the right)
• Increases gradually, growing more slowly as $x$ gets larger
• Looks like half of a sideways parabola

Key points: $(0, 0)$, $(1, 1)$, $(4, 2)$, $(9, 3)$

The basic cube root function $f(x) = \sqrt[3]{x}$:

• Passes through the origin (0, 0)
• Exists for all real numbers (extends left and right)
• Increases throughout, but more steeply near the origin
• Has a distinctive S-shaped curve

Key points: $(-8, -2)$, $(-1, -1)$, $(0, 0)$, $(1, 1)$, $(8, 2)$

Transformations of Radical Functions

Just like with other functions, you can shift, stretch, compress, and reflect radical functions. The same transformation rules apply.

For a function in the form $f(x) = a\sqrt[n]{x - h} + k$:

• $h$ shifts the graph horizontally: right if $h > 0$, left if $h < 0$
• $k$ shifts the graph vertically: up if $k > 0$, down if $k < 0$
• $a$ stretches or compresses vertically: if $|a| > 1$ it stretches; if $|a| < 1$ it compresses
• Negative $a$ reflects the graph over the $x$-axis

For example, $f(x) = -\sqrt[3]{x + 2} + 1$ takes the basic cube root function and:

1. Shifts it 2 units left (because of $x + 2$, which is $x - (-2)$)
2. Reflects it over the $x$-axis (because of the negative sign)
3. Shifts it 1 unit up (because of $+ 1$)

The Secret Connection: Rational Exponents

Here is the beautiful insight that ties everything together. Radicals and exponents are not separate topics; they are two notations for the same thing.

$$\sqrt[n]{a} = a^{1/n}$$

This means the $n$th root of $a$ is the same as $a$ raised to the power $1/n$. For example:

• $\sqrt{9} = 9^{1/2} = 3$
• $\sqrt[3]{8} = 8^{1/3} = 2$
• $\sqrt[4]{16} = 16^{1/4} = 2$

But it goes further. What about $a^{2/3}$? That means:

$$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$

The denominator of the exponent tells you the root; the numerator tells you the power. You can do them in either order.

For example, $8^{2/3}$ can be calculated as:

• $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$, or
• $\left(\sqrt[3]{8}\right)^2 = 2^2 = 4$

Either way, you get the same answer. Usually, taking the root first (second method) involves smaller numbers and is easier.

Why does this matter? Because all those exponent rules you learned now apply to radicals:

$$a^{m} \cdot a^{n} = a^{m+n}$$ $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ $$(a^{m})^{n} = a^{mn}$$

These rules work with fractional exponents too, making simplification much more systematic.

Simplifying Expressions with Rational Exponents

When you need to simplify expressions involving rational exponents, use the same rules you know from working with integer exponents.

For example, to simplify $\frac{x^{2/3} \cdot x^{1/2}}{x^{-1/6}}$:

First, handle the numerator using the product rule: $$x^{2/3} \cdot x^{1/2} = x^{2/3 + 1/2}$$

To add these fractions, find a common denominator (6): $$\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$$

So the numerator is $x^{7/6}$.

Now divide by $x^{-1/6}$ using the quotient rule: $$\frac{x^{7/6}}{x^{-1/6}} = x^{7/6 - (-1/6)} = x^{7/6 + 1/6} = x^{8/6} = x^{4/3}$$

You can leave the answer as $x^{4/3}$ or write it as $\sqrt[3]{x^4}$. Both are correct.

Solving Radical Equations

A radical equation is an equation where the variable appears under a radical sign. To solve these, you need to eliminate the radical by raising both sides to the appropriate power.

The general process:

1. Isolate the radical on one side of the equation
2. Raise both sides to the power that matches the index (square both sides for square roots, cube both sides for cube roots, etc.)
3. Solve the resulting equation
4. Check your answers in the original equation

For a square root equation like $\sqrt{3x + 1} = 4$:

1. The radical is already isolated
2. Square both sides: $3x + 1 = 16$
3. Solve: $3x = 15$, so $x = 5$
4. Check: $\sqrt{3(5) + 1} = \sqrt{16} = 4$ confirms the answer

Extraneous Solutions: The Trap You Must Avoid

When you raise both sides of an equation to a power, you can accidentally create solutions that do not actually work in the original equation. These false solutions are called extraneous solutions.

Why does this happen? Consider this: if $a = b$, then certainly $a^2 = b^2$. But the reverse is not always true. If $a^2 = b^2$, then $a$ could equal $b$ or $a$ could equal $-b$.

This is why checking your answers is not optional when solving radical equations. It is a required step.

Consider the equation $\sqrt{x + 5} = x - 1$. After squaring and solving, you might get two candidate solutions. But when you check each one in the original equation, only the solutions that make both sides equal are valid. Any solution where the left side (which involves a square root) would need to equal a negative right side is automatically extraneous, since square roots cannot be negative.

Notation and Terminology

Term	Meaning	Example
$\sqrt[n]{x}$	$n$th root of $x$	$\sqrt[3]{8} = 2$
$a^{1/n}$	Same as $\sqrt[n]{a}$	$27^{1/3} = 3$
$a^{m/n}$	Same as $\sqrt[n]{a^m}$ or $(\sqrt[n]{a})^m$	$8^{2/3} = 4$
Index	The $n$ in $\sqrt[n]{x}$; indicates which root	In $\sqrt[3]{8}$, the index is 3
Radicand	The expression under the radical	In $\sqrt{5x+2}$, the radicand is $5x+2$
Radical function	Function containing a radical	$f(x) = \sqrt{x-3}$
Rational exponent	An exponent that is a fraction	$x^{3/4}$
Radical equation	Equation with variable under radical	$\sqrt{x + 3} = 5$
Extraneous solution	“Solution” from algebra that does not satisfy the original	Must always check

Examples

Key Properties and Rules

Rational Exponent Rules

All the exponent rules work with fractional exponents:

Product Rule: $$a^{m/n} \cdot a^{p/q} = a^{m/n + p/q}$$

Quotient Rule: $$\frac{a^{m/n}}{a^{p/q}} = a^{m/n - p/q}$$

Power Rule: $$(a^{m/n})^{p/q} = a^{(m/n)(p/q)} = a^{mp/nq}$$

Converting Between Forms: $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Domain Rules for Radical Functions

Root Type	Example	Domain
Even index ($n = 2, 4, 6, …$)	$\sqrt{x}$, $\sqrt[4]{x}$	Radicand must be $\geq 0$
Odd index ($n = 3, 5, 7, …$)	$\sqrt[3]{x}$, $\sqrt[5]{x}$	All real numbers

Transformation Rules for $f(x) = a\sqrt[n]{x - h} + k$

Parameter	Effect
$h > 0$	Shifts right $h$ units
$h < 0$	Shifts left $
$k > 0$	Shifts up $k$ units
$k < 0$	Shifts down $
$	a
$0 <	a
$a < 0$	Reflection over $x$-axis

Steps for Solving Radical Equations

1. Isolate the radical on one side
2. Raise both sides to the power of the index
3. Solve the resulting equation
4. Check all solutions in the original equation
5. Reject any extraneous solutions

Real-World Applications

Pendulum Period

The time it takes a pendulum to complete one full swing (called the period) follows this formula:

$$T = 2\pi\sqrt{\frac{L}{g}}$$

where $L$ is the length of the pendulum and $g \approx 9.8 \text{ m/s}^2$ is the acceleration due to gravity. Notice that this is a radical function of $L$. Doubling the length does not double the period; instead, it multiplies the period by $\sqrt{2} \approx 1.414$. This is why grandfather clocks need careful calibration: small changes in pendulum length cause noticeable changes in timekeeping.

Escape Velocity

The minimum speed needed for an object to escape a planet’s gravitational pull is:

$$v = \sqrt{\frac{2GM}{r}}$$

where $G$ is the gravitational constant, $M$ is the planet’s mass, and $r$ is the distance from the planet’s center. For Earth, this works out to about 11.2 km/s (roughly 25,000 mph). The square root means that even a planet four times as massive only requires twice the escape velocity.

Distance Formula

The distance between two points in a coordinate plane uses a square root:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

This comes directly from the Pythagorean theorem. In three dimensions, you just add another term:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Kepler’s Third Law (Planetary Motion)

Johannes Kepler discovered that the orbital period of a planet is related to its distance from the Sun by:

$$T = k \cdot r^{3/2}$$

where $T$ is the orbital period, $r$ is the average distance from the Sun, and $k$ is a constant. The exponent $3/2$ means you take the distance, cube it, and then take the square root. This elegant relationship governs everything from Mercury’s 88-day orbit to Neptune’s 165-year journey around the Sun.

Self-Test Problems

Summary

• Radical functions have the form $f(x) = \sqrt[n]{x}$, where $n$ is the index (which root you are taking)
• Even roots (square root, fourth root, etc.) require non-negative radicands; odd roots (cube root, fifth root, etc.) accept any real number
• Rational exponents provide an alternative notation: $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
• All exponent rules work with fractional exponents: product rule, quotient rule, and power rule
• Transformations of radical functions follow the same patterns as other functions: $f(x) = a\sqrt[n]{x - h} + k$ shifts, stretches, and reflects the basic curve
• To solve radical equations: isolate the radical, raise both sides to the power of the index, solve, then check all solutions
• Extraneous solutions can appear when you raise both sides to a power. Always substitute back into the original equation to verify
• Radical functions appear throughout science and engineering: pendulum periods, escape velocity, distance calculations, and orbital mechanics all use these concepts