Numbers and Operations

Radicals

Properties of radicals can be used to simplify radical expressions.
The square root of a number is a number that is multiplied by itself to give that number. For example, the square root of 9 is 3 because:
33=93\cdot3=9
The nnth root of a number is multiplied by itself nn times to get that number. For example, the fourth root of 81 is 3 because:
3333=813\cdot3\cdot3\cdot3=81
A radical symbol 0\sqrt{\phantom0} is used to represent nnth roots, where nn is a natural number greater than or equal to 2. In the radical xn\sqrt[n]{x}, the index is nn, which represents the nnth root of xx. For example, the index of x3\sqrt[3]{x} is 3. In the radical x\sqrt{x}, the index is not shown, but it is understood to be 2. The radicand is the expression under the radical.
  • In the radical x+2\sqrt{x+2}, the index is 2, and the radicand is x+2x+2. It is read as "the square root of the quantity xx plus 2."
  • In the radical x23\sqrt[3]{x^2} , the index is 3, and the radicand is x2x^2. It is read as "the cube root of xx squared."
A radical with index nn is the inverse, or opposite operation, of the exponent nn. If nn is even and x0x\geq0 or if nn is odd:
xnn=(xn)n=x\sqrt[n]{x^n}=(\sqrt[n]{x})^n=x
If nn is even and x<0x<0, then xx\sqrt[x]{x} is not a real number. For example, (4)33=4\sqrt[3]{(-4)^3}=-4, and 4\sqrt{-4} is not a real number.

The rules for multiplying or dividing radicals are the same as with exponents. A product or quotient under a radical can be split into two separate radicals, and radicals with the same index that are multiplied or divided can be combined into one.

Properties of radicals are used to simplify radical expressions. It is also possible to combine like radicals, which have the same radicand and index.

Multiplying and Dividing Radicals

Property Example Description
Product of radicals property
xyn=xnyn\sqrt[n]{xy}=\sqrt[n]{x}\sqrt[n]{y}
The radical of a product is the product of the radicals.
Quotient of radicals property
xyn=xnyn\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}
The radical of a quotient is the quotient of the radicals.

Step-By-Step Example
Simplifying Radical Expressions
Simplify the expression.
23+36\sqrt{2}\sqrt{3}+3\sqrt{6}
Step 1
Use the product of radicals property to combine 2\sqrt{2} and 3\sqrt{3}.
23+3623+36\begin{gathered}\sqrt{2}\sqrt{3}+3\sqrt{6}\\\sqrt{2\cdot3}+3\sqrt{6}\\\end{gathered}
Step 2
Simplify the expression. Multiply 2 and 3 inside the radical.
23+366+36\begin{gathered}\sqrt{2\cdot3}+3\sqrt{6}\\\sqrt{6}+3\sqrt{6}\end{gathered}
Solution
Combine like radicals. The terms 6\sqrt{6} and 363\sqrt{6} have the same radicand and index. So, they can be combined like other like terms by adding the coefficients. The coefficient of 6\sqrt{6} is 1.
6+3646\begin{gathered}\sqrt{6}+3\sqrt{6}\\4\sqrt{6}\end{gathered}

Simplest Radical Form

A radical expression in simplest form has no perfect powers or fractions in the radicand and no fractions in the denominator of a fraction.
When simplifying radicals, there is a set of rules that determines whether the radical is in simplest form.

Rules for Simplifying Radicals

Rule Example
There are no perfect nnth powers under a radical of index nn, such as perfect squares within a square root or perfect cubes under a cube root.
8=(4)(2)=42=22\begin{aligned}\sqrt{8}&=\sqrt{(4)(2)}\\&=\sqrt{4}\sqrt{2}\\&=2\sqrt{2}\end{aligned}
There are no fractions under a radical or radicals in the denominator of a fraction.
34=34=32\begin{aligned}\sqrt{\frac{3}{4}}&=\frac{\sqrt{3}}{\sqrt{4}}\\&=\frac{\sqrt{3}}{2}\end{aligned}

Step-By-Step Example
Writing Radicals in Simplest Form
Simplify x74\sqrt[4]{x^7} for x0x\geq0.
Step 1
The index of the radical is 4. Rewrite x7x^7 as a product of a perfect fourth power and another power.
x74x4x34\begin{gathered}\sqrt[4]{x^7}\\\sqrt[4]{x^4x^3}\end{gathered}
Step 2
Use the product of radicals property to split the radical into a product.
x4x34x44x34\begin{gathered}\sqrt[4]{x^4x^3}\\\sqrt[4]{x^4}\sqrt[4]{x^3}\end{gathered}
Solution
Use the inverse relationship between the radical and exponent to rewrite x44\sqrt[4]{x^4}. This expression is equal to xx because x0x\geq0.
x44x34xx34\begin{gathered}\sqrt[4]{x^4}\sqrt[4]{x^3}\\x\sqrt[4]{x^3}\end{gathered}

Rationalizing the Denominator

Fractions with radicals in the denominator can be simplified by multiplying by a fraction equivalent to 1.

Rationalizing the denominator is the process of rewriting a fraction to remove radical expressions from the denominator. To rationalize the denominator, multiply the fraction by a fraction that is equivalent to 1. The type of radical expression in the denominator determines which fraction equivalent to 1 should be used.

If the denominator has two terms, such as x+y\sqrt{x}+y, the conjugate of the expression is xy\sqrt{x}-y. When two expressions are multiplied, the resulting expression does not contain a radical:
(x+y)(xy)(x)2yx+yxy2xy2\begin{gathered}\left(\sqrt{x}+y\right)\left(\sqrt{x}-y\right)\\\left(\sqrt{x}\right)^2-y\sqrt{x}+y\sqrt{x}-y^2\\x-y^2\end{gathered}

How to Rationalize Denominators

Denominator Fraction Equivalent to 1 Example
Single square root Multiply by a fraction in this form:
xx=1\frac{\sqrt{x}}{\sqrt{x}}=1
25(55)=2525=255\begin{aligned}\frac{2}{\sqrt{5}}\left(\frac{\sqrt{5}}{\sqrt{5}}\right)&=\frac{2\sqrt{5}}{\sqrt{25}} \\&= \frac{2\sqrt{5}}{5}\end{aligned}
Single cube root Multiply by a fraction in this form:
x23x23=1\frac{\sqrt[3]{x^2}}{\sqrt[3]{x^2}}=1
253(523523)=2523533=25235\begin{aligned}\frac{2}{\sqrt[3]{5}}\left(\frac{\sqrt[3]{5^2}}{\sqrt[3]{5^2}}\right)&=\frac{2\sqrt[3]{5^2}}{\sqrt[3]{5^3}}\\&=\frac{2\sqrt[3]{5^2}}{5}\end{aligned}
Expression with two terms, such as:
x+y\sqrt{x}+y
Multiply by a fraction in this form:
xyxy=1\frac{\sqrt{x}-y}{\sqrt{x}-y}=1
25+1(5151)=252255+51=2524\begin{aligned}\frac{2}{\sqrt{5}+1}\left(\frac{\sqrt{5}-1}{\sqrt{5}-1}\right)&=\frac{2\sqrt{5}-2}{\sqrt{25}-\sqrt{5}+\sqrt{5}-1}\\&=\frac{2\sqrt{5}-2}{4}\end{aligned}

Radicals and Rational Exponents

Rational exponents can be used to represent radicals, or nnth roots.
Radicals can be expressed by using rational exponents. A rational exponent is an exponent that is a rational number. If the exponent is written as a fraction ab\frac{a}{b}, then the denominator bb is the index of the radical, and the numerator aa is the exponent of the radicand. Alternately, the expression can be written in two ways, with the entire radical raised to the aa power:
xab=(xa)1b=xab=(xb)ax^{\frac{a}{b}}=(x^a)^\frac{1}{b}=\sqrt[b]{x^a}=(\sqrt[b]{x})^a
xab=(x1b)a=xab=(xb)ax^{\frac{a}{b}}=(x^\frac{1}{b})^a=\sqrt[b]{x^a}=(\sqrt[b]{x})^a
A numerator of 1 indicates the first power:
x1b=xbx^{\frac{1}{b}}=\sqrt[b]{x}
A denominator of 2 indicates a square root:
x12=xx^{\frac{1}{2}}=\sqrt{x}
Note that rational exponents can be positive or negative.
To simplify expressions with rational exponents, rewrite the expression in radical form.