Lesson 1.3: Radicals and Rational Exponents

New Concept

This lesson introduces radicals and rational exponents, the tools for finding roots of numbers. We'll start with the principal square root, $\sqrt{a}$, and build rules to simplify, add, subtract, and rationalize various radical expressions.

What’s next

This card is just the beginning. Next, you'll tackle interactive examples and practice cards to master simplifying square roots, rationalizing denominators, and using rational exponents.

Property

The principal square root of $a$ is the nonnegative number that, when multiplied by itself, equals $a$. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $\sqrt{a}$.

Examples

• To evaluate $\sqrt{144}$, we find the number that squares to 144. Since $12^2 = 144$, $\sqrt{144} = 12$.
• To evaluate $\sqrt{49 + 576}$, we first perform the addition inside the radicand: $\sqrt{625}$. Since $25^2 = 625$, the result is 25.
• To evaluate $\sqrt{100} - \sqrt{64}$, we find each square root separately: $10 - 8$. The result is 2.

Explanation

Even though both $5^2$ and $(-5)^2$ equal 25, the radical symbol $\sqrt{a}$ specifically asks for the positive root, known as the principal root. So, $\sqrt{25}$ is just 5, not $\pm 5$.

Property

If $a$ and $b$ are nonnegative, the square root of the product $ab$ is equal to the product of the square roots of $a$ and $b$. To simplify, factor any perfect squares from the radicand, write the radical as a product of separate radical expressions, and simplify each one.

Examples

• To simplify $\sqrt{200}$, factor it into a perfect square and another number: $\sqrt{100 \cdot 2}$. This becomes $\sqrt{100} \cdot \sqrt{2}$, which simplifies to $10\sqrt{2}$.
• To simplify $\sqrt{72x^2y^4}$, factor out perfect squares: $\sqrt{36x^2y^4 \cdot 2}$. This separates into $\sqrt{36x^2y^4} \cdot \sqrt{2}$, giving $6xy^2\sqrt{2}$.
• To simplify $\sqrt{8} \cdot \sqrt{2}$, we can combine them under one radical using the product rule: $\sqrt{8 \cdot 2} = \sqrt{16}$, which simplifies to 4.

Explanation

This rule allows you to break down a large radical into smaller, more manageable parts. By factoring out perfect squares (like 4, 9, 25), you can simplify the expression piece by piece, making complex roots easier to solve.

Property

The square root of the quotient $\frac{a}{b}$ is equal to the quotient of the square roots of $a$ and $b$, where $b \neq 0$. To use this rule, write the radical expression as a quotient of two separate radical expressions, then simplify the numerator and denominator.

Examples

• To simplify $\sqrt{\frac{7}{64}}$, apply the quotient rule to get $\frac{\sqrt{7}}{\sqrt{64}}$. This simplifies to $\frac{\sqrt{7}}{8}$.
• To simplify $\sqrt{\frac{3m^2}{49n^4}}$, separate the terms: $\frac{\sqrt{3m^2}}{\sqrt{49n^4}}$. This simplifies to $\frac{m\sqrt{3}}{7n^2}$.
• To simplify $\frac{\sqrt{150a^9b}}{\sqrt{6a^5b}}$, combine them into one fraction under the radical: $\sqrt{\frac{150a^9b}{6a^5b}} = \sqrt{25a^4}$. This simplifies to $5a^2$.

Explanation

This rule lets you split a fraction inside a square root into two separate problems: one for the numerator and one for the denominator. This makes simplifying fractions under a radical much more straightforward.

Property

We can add or subtract radical expressions only when they have the same radicand and the same radical type. To perform addition or subtraction:

1. Simplify each radical expression completely.
2. Add or subtract the coefficients of the terms with equal radicands.

Examples

• To add $3\sqrt{27} + 4\sqrt{3}$, first simplify $\sqrt{27}$ to $3\sqrt{3}$. The expression becomes $3(3\sqrt{3}) + 4\sqrt{3} = 9\sqrt{3} + 4\sqrt{3} = 13\sqrt{3}$.
• To subtract $10\sqrt{50x^3} - 3\sqrt{8x^3}$, simplify both terms: $10(5x\sqrt{2x}) - 3(2x\sqrt{2x}) = 50x\sqrt{2x} - 6x\sqrt{2x} = 44x\sqrt{2x}$.
• To simplify $\sqrt{20} + \sqrt{45} - \sqrt{5}$, simplify each term to get $2\sqrt{5} + 3\sqrt{5} - \sqrt{5}$. Combining these like terms gives $4\sqrt{5}$.

Explanation

Think of radicals with the same radicand as like terms. Just as you can add $3x + 4x$ to get $7x$, you can add $3\sqrt{5} + 4\sqrt{5}$ to get $7\sqrt{5}$. If the radicands are different, you cannot combine them.

Property

To write an expression in simplest form, it must not contain a radical in the denominator. This is done by multiplying the numerator and denominator by a form of 1 that eliminates the radical.

• If the denominator is a single term like $b\sqrt{c}$, multiply by $\frac{\sqrt{c}}{\sqrt{c}}$.
• If the denominator is a sum or difference like $a + b\sqrt{c}$, multiply by its conjugate, $a - b\sqrt{c}$.

Examples

• To rationalize $\frac{5\sqrt{2}}{4\sqrt{7}}$, multiply by $\frac{\sqrt{7}}{\sqrt{7}}$ to get $\frac{5\sqrt{14}}{4 \cdot 7} = \frac{5\sqrt{14}}{28}$.
• To rationalize $\frac{6}{2+\sqrt{7}}$, multiply by the conjugate $\frac{2-\sqrt{7}}{2-\sqrt{7}}$. This yields $\frac{6(2-\sqrt{7})}{4-7} = \frac{12-6\sqrt{7}}{-3} = 2\sqrt{7}-4$.
• To rationalize $\frac{10}{\sqrt{6}-1}$, multiply by the conjugate $\frac{\sqrt{6}+1}{\sqrt{6}+1}$. This gives $\frac{10(\sqrt{6}+1)}{6-1} = \frac{10(\sqrt{6}+1)}{5} = 2(\sqrt{6}+1) = 2\sqrt{6}+2$.

Explanation

Rationalizing is a technique to move a radical from the denominator to the numerator without changing the fraction's value. This is considered a best practice for writing expressions in their simplest, standard form.

Property

If $a$ is a real number with at least one $n$th root, then the principal $n$th root of $a$, written as $\sqrt[n]{a}$, is the number with the same sign as $a$ that, when raised to the $n$th power, equals $a$. The integer $n$ is called the index of the radical.

Examples

• To simplify $\sqrt[3]{-64}$, we look for a number that, when cubed, gives -64. Since $(-4)^3 = -64$, the answer is -4.
• To simplify $\sqrt[4]{16 \cdot 81}$, we can use the product rule for roots: $\sqrt[4]{16} \cdot \sqrt[4]{81}$. This gives $2 \cdot 3 = 6$.
• To simplify $5\sqrt[3]{16} - \sqrt[3]{2}$, first simplify $\sqrt[3]{16}$ to $\sqrt[3]{8 \cdot 2} = 2\sqrt[3]{2}$. The expression becomes $5(2\sqrt[3]{2}) - \sqrt[3]{2} = 10\sqrt[3]{2} - \sqrt[3]{2} = 9\sqrt[3]{2}$.

Explanation

This extends the idea of a square root to any root. The index $n$ tells you how many times a number must be multiplied by itself to equal the radicand. For example, $\sqrt[3]{8}$ asks, 'What number cubed equals 8?'

Property

Rational exponents are another way to express principal $n$th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

Examples

• To simplify $64^{\frac{2}{3}}$, we can write it as $(\sqrt[3]{64})^2$. The cube root of 64 is 4, and $4^2$ is 16.
• To write $\frac{5}{\sqrt[4]{x^3}}$ using a rational exponent, first convert the radical to $x^{\frac{3}{4}}$. The expression is $\frac{5}{x^{\frac{3}{4}}}$, which can be written as $5x^{-\frac{3}{4}}$.
• To simplify $(27x^9)^{\frac{2}{3}}$, apply the exponent to both parts: $27^{\frac{2}{3}} \cdot (x^9)^{\frac{2}{3}}$. This becomes $(\sqrt[3]{27})^2 \cdot x^{9 \cdot \frac{2}{3}} = 3^2 \cdot x^6 = 9x^6$.

Explanation

A fractional exponent is a compact way to write a radical. The denominator of the fraction represents the root (the index), and the numerator represents the power. It is often easiest to find the root first, then apply the power.