More Basic Functions and Graphs: Background You’ll Need 2

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More Basic Functions and Graphs: Background You’ll Need 2

Convert between radical and rational exponent notations

Radical and Rational Exponent Notations

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as

\begin{array}{ccc} \sqrt{3 y + 18} = x \\ \sqrt{x + 3} = y - 3 \\ \sqrt{x + 5} - \sqrt{y - 3} = 2 \end{array}

$\begin{array}{ccc} \sqrt{3y+18}=x & \\ \sqrt{x+3}=y-3 & \\ \sqrt{x+5}-\sqrt{y - 3}=2\end{array}$

Radical equations are manipulated by eliminating each radical, one at a time until you have solved for the indicated variable.

radical equation

An equation containing terms with a variable in the radicand is called a radical equation.

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, ${16}^{\frac{1}{2}}$ is another way of writing $\sqrt{16}$ and ${8}^{\frac{2}{3}}$ is another way of writing $\left(\sqrt[3]{8}\right)^2$ .

We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals $1$ .

rational exponent

A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:

a^{\frac{m}{n}} = {(a^{\frac{1}{n}})}^{m} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(\sqrt[n]{a})}^{m}

${a}^{\frac{m}{n}}={\left({a}^{\frac{1}{n}}\right)}^{m}={\left({a}^{m}\right)}^{\frac{1}{n}}=\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}$

To convert a radical to an exponent notation, remember that the $n$ th root of a number can be written as a power with a fractional exponent. The denominator of the fraction is the root’s index ( $n$ ), and the numerator is the power to which the radicand is raised.

The square root of

a

$a$ , written as

\sqrt{a}

$\sqrt{a}$ , can be expressed as

a^{\frac{1}{2}}

$a^{\frac{1}{2}}$ .
The cube root of

a

$a$ , written as

\sqrt[3]{a}

$\sqrt[3]{a}$ , can be expressed as

a^{\frac{1}{3}}

$a^{\frac{1}{3}}$ .

Conversely, to convert an expression from exponent notation to radical notation, use the denominator of the exponent’s fraction as the index of the radical, and the numerator as the power inside the radical.

a^{\frac{3}{2}}

$a^{\frac{3}{2}}$ can be written as

\sqrt[2]{a^{3}}

$\sqrt[2]{a^3}$ or

\sqrt[3]{a^{3}}

$\sqrt[3]{a^3}$ .

a^{\frac{2}{5}}

$a^{\frac{2}{5}}$ can be written as

\sqrt[5]{a^{2}}

$\sqrt[5]{a^2}$ .

Convert the fifth root of

x^{3}

$x^3$ , written as

\sqrt[5]{a^{3}}

$\sqrt[5]{a^3}$ , to exponent notation.

Identify the index of the root, which is $5$ in this case.
Write the radicand (the number under the radical) with an exponent that represents the power it is raised to, which is 3 here.
Combine the root and the power into a single exponent using the rule ${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}$
Apply the rule to get $x^{\frac{3}{5}}$ .

The exponent notation for $\sqrt[5]{a^3}$ is $x^{\frac{3}{5}}$ .

Convert

y^{\frac{4}{3}}

$y^{\frac{4}{3}}$ into radical notation.

The denominator of the fraction in the exponent ( $3$ ) is the index of the radical.
The numerator ( $4$ ) will be the power to which the radicand is raised inside the radical.
Write the radical with the index and raise the radicand to the power of the numerator.
The conversion is then written as $\sqrt[3]{y^4}$ .

The radical notation for $y^{\frac{4}{3}}$ is $\sqrt[3]{y^4}$ .