Rational exponents and radicals are two different ways to write the same mathematical expression. In \(x^{a/b}\), the denominator \(b\) tells us the root and the numerator \(a\) tells us the power. The problems below practice converting between rational exponent form and radical form in both directions.
Lesson
Notes
Rational Exponent Form & Radical Form
\(\displaystyle x^{a/b} = \sqrt[b]{x^a} = \left(\sqrt[b]{x}\right)^a\)
Practice Problems & Videos
\(\textbf{1)}\) Express \(\sqrt[3]{x^2}\) in Rational Exponent Form The answer is \(\displaystyle x^{2/3}\)
\(\,\,\,\,\,\,\sqrt[3]{x^2}\)
\(\,\,\,\,\,\,\displaystyle\left(x^2\right)^{1/3}\)
\(\,\,\,\,\,\,\displaystyle x^{2/3}\)
The answer is \(\displaystyle x^{2/3}\)
\(\,\,\,\,\,\,\sqrt[3]{x^2}\)
\(\,\,\,\,\,\,\displaystyle\left(x^2\right)^{1/3}\)
\(\,\,\,\,\,\,\displaystyle x^{2/3}\)
The answer is \(\displaystyle x^{2/3}\)
\(\textbf{2)}\) Express \(\sqrt[5]{2^3}\) in Rational Exponent Form The answer is \(\displaystyle 2^{3/5}\)
\(\,\,\,\,\,\,\sqrt[5]{2^3}\)
\(\,\,\,\,\,\,\displaystyle\left(2^3\right)^{1/5}\)
\(\,\,\,\,\,\,\displaystyle2^{3/5}\)
The answer is \(\displaystyle2^{3/5}\)
\(\,\,\,\,\,\,\sqrt[5]{2^3}\)
\(\,\,\,\,\,\,\displaystyle\left(2^3\right)^{1/5}\)
\(\,\,\,\,\,\,\displaystyle2^{3/5}\)
The answer is \(\displaystyle2^{3/5}\)
\(\textbf{3)}\) Express \(\sqrt[4]{3.1^7}\) in Rational Exponent Form The answer is \(\displaystyle 3.1^{7/4}\)
\(\,\,\,\,\,\,\sqrt[4]{3.1^7}\)
\(\,\,\,\,\,\,\displaystyle\left(3.1^7\right)^{1/4}\)
\(\,\,\,\,\,\,\displaystyle3.1^{7/4}\)
The answer is \(\displaystyle3.1^{7/4}\)
\(\,\,\,\,\,\,\sqrt[4]{3.1^7}\)
\(\,\,\,\,\,\,\displaystyle\left(3.1^7\right)^{1/4}\)
\(\,\,\,\,\,\,\displaystyle3.1^{7/4}\)
The answer is \(\displaystyle3.1^{7/4}\)
\(\textbf{4)}\) Express \(\sqrt[5]{x^3}\) in Rational Exponent Form The answer is \(\displaystyle x^{3/5}\)
\(\,\,\,\,\,\,\sqrt[5]{x^3}\)
\(\,\,\,\,\,\,\displaystyle\left(x^3\right)^{1/5}\)
\(\,\,\,\,\,\,\displaystyle x^{3/5}\)
The answer is \(\displaystyle x^{3/5}\)
\(\,\,\,\,\,\,\sqrt[5]{x^3}\)
\(\,\,\,\,\,\,\displaystyle\left(x^3\right)^{1/5}\)
\(\,\,\,\,\,\,\displaystyle x^{3/5}\)
The answer is \(\displaystyle x^{3/5}\)
\(\textbf{5)}\) Express \(\displaystyle z^{1/7}\) in Radical Form The answer is \(\sqrt[7]{z}\)
\(\,\,\,\,\,\,\displaystyle z^{1/7}\)
\(\,\,\,\,\,\,\displaystyle \sqrt[7]{z^1}\)
\(\,\,\,\,\,\,\displaystyle \sqrt[7]{z}\)
The answer is \(\sqrt[7]{z}\)
\(\,\,\,\,\,\,\displaystyle z^{1/7}\)
\(\,\,\,\,\,\,\displaystyle \sqrt[7]{z^1}\)
\(\,\,\,\,\,\,\displaystyle \sqrt[7]{z}\)
The answer is \(\sqrt[7]{z}\)
\(\textbf{6)}\) Express \(\displaystyle 5^{5/2}\) in Radical Form The answer is \(\sqrt{5^5}\) or \(\left(\sqrt{5}\right)^5\)
\(\,\,\,\,\,\,\displaystyle5^{5/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[2]{5^5}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{5^5}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt{5}\right)^5\)
The answer is \(\sqrt{5^5}\) or \(\left(\sqrt{5}\right)^5\)
\(\,\,\,\,\,\,\displaystyle5^{5/2}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[2]{5^5}\)
\(\,\,\,\,\,\,\displaystyle\sqrt{5^5}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt{5}\right)^5\)
The answer is \(\sqrt{5^5}\) or \(\left(\sqrt{5}\right)^5\)
\(\textbf{7)}\) Express \(\displaystyle 4.2^{3/4}\) in Radical Form The answer is \(\sqrt[4]{4.2^3}\) or \(\left(\sqrt[4]{4.2}\right)^3\)
\(\,\,\,\,\,\,\displaystyle4.2^{3/4}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{4.2^3}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[4]{4.2}\right)^3\)
The answer is \(\sqrt[4]{4.2^3}\) or \(\left(\sqrt[4]{4.2}\right)^3\)
\(\,\,\,\,\,\,\displaystyle4.2^{3/4}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{4.2^3}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[4]{4.2}\right)^3\)
The answer is \(\sqrt[4]{4.2^3}\) or \(\left(\sqrt[4]{4.2}\right)^3\)
\(\textbf{8)}\) Express \(\displaystyle x^{1/4}\) in Radical Form The answer is \(\sqrt[4]{x}\)
\(\,\,\,\,\,\,\displaystyle x^{1/4}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{x^1}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{x}\)
The answer is \(\sqrt[4]{x}\)
\(\,\,\,\,\,\,\displaystyle x^{1/4}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{x^1}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[4]{x}\)
The answer is \(\sqrt[4]{x}\)
\(\textbf{9)}\) Express \(\displaystyle x^{2/3}\) in Radical Form The answer is \(\sqrt[3]{x^2}\) or \(\left(\sqrt[3]{x}\right)^2\)
\(\,\,\,\,\,\,\displaystyle x^{2/3}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[3]{x^2}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[3]{x}\right)^2\)
The answer is \(\sqrt[3]{x^2}\) or \(\left(\sqrt[3]{x}\right)^2\)
\(\,\,\,\,\,\,\displaystyle x^{2/3}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[3]{x^2}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[3]{x}\right)^2\)
The answer is \(\sqrt[3]{x^2}\) or \(\left(\sqrt[3]{x}\right)^2\)
\(\textbf{10)}\) Express \(\displaystyle 3^{4/5}\) in Radical Form The answer is \(\sqrt[5]{3^4}\) or \(\left(\sqrt[5]{3}\right)^4\)
\(\,\,\,\,\,\,\displaystyle3^{4/5}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[5]{3^4}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[5]{3}\right)^4\)
The answer is \(\sqrt[5]{3^4}\) or \(\left(\sqrt[5]{3}\right)^4\)
\(\,\,\,\,\,\,\displaystyle3^{4/5}\)
\(\,\,\,\,\,\,\displaystyle\sqrt[5]{3^4}\)
\(\,\,\,\,\,\,\displaystyle\left(\sqrt[5]{3}\right)^4\)
The answer is \(\sqrt[5]{3^4}\) or \(\left(\sqrt[5]{3}\right)^4\)
\(\textbf{11)}\) Express \(\sqrt[6]{y^5}\) in Rational Exponent Form The answer is \(\displaystyle y^{5/6}\)
\(\,\,\,\,\,\,\sqrt[6]{y^5}\)
\(\,\,\,\,\,\,\displaystyle\left(y^5\right)^{1/6}\)
\(\,\,\,\,\,\,\displaystyle y^{5/6}\)
The answer is \(\displaystyle y^{5/6}\)
\(\,\,\,\,\,\,\sqrt[6]{y^5}\)
\(\,\,\,\,\,\,\displaystyle\left(y^5\right)^{1/6}\)
\(\,\,\,\,\,\,\displaystyle y^{5/6}\)
The answer is \(\displaystyle y^{5/6}\)
\(\textbf{12)}\) Express \(\sqrt[3]{m^7}\) in Rational Exponent Form The answer is \(\displaystyle m^{7/3}\)
\(\,\,\,\,\,\,\sqrt[3]{m^7}\)
\(\,\,\,\,\,\,\displaystyle\left(m^7\right)^{1/3}\)
\(\,\,\,\,\,\,\displaystyle m^{7/3}\)
The answer is \(\displaystyle m^{7/3}\)
\(\,\,\,\,\,\,\sqrt[3]{m^7}\)
\(\,\,\,\,\,\,\displaystyle\left(m^7\right)^{1/3}\)
\(\,\,\,\,\,\,\displaystyle m^{7/3}\)
The answer is \(\displaystyle m^{7/3}\)
\(\textbf{13)}\) Express \(\sqrt[8]{a^3}\) in Rational Exponent Form The answer is \(\displaystyle a^{3/8}\)
\(\,\,\,\,\,\,\sqrt[8]{a^3}\)
\(\,\,\,\,\,\,\displaystyle\left(a^3\right)^{1/8}\)
\(\,\,\,\,\,\,\displaystyle a^{3/8}\)
The answer is \(\displaystyle a^{3/8}\)
\(\,\,\,\,\,\,\sqrt[8]{a^3}\)
\(\,\,\,\,\,\,\displaystyle\left(a^3\right)^{1/8}\)
\(\,\,\,\,\,\,\displaystyle a^{3/8}\)
The answer is \(\displaystyle a^{3/8}\)
\(\textbf{14)}\) Express \(\left(\sqrt[5]{x}\right)^4\) in Rational Exponent Form The answer is \(\displaystyle x^{4/5}\)
\(\,\,\,\,\,\,\left(\sqrt[5]{x}\right)^4\)
\(\,\,\,\,\,\,\left(x^{1/5}\right)^4\)
\(\,\,\,\,\,\,x^{4/5}\)
The answer is \(\displaystyle x^{4/5}\)
\(\,\,\,\,\,\,\left(\sqrt[5]{x}\right)^4\)
\(\,\,\,\,\,\,\left(x^{1/5}\right)^4\)
\(\,\,\,\,\,\,x^{4/5}\)
The answer is \(\displaystyle x^{4/5}\)
\(\textbf{15)}\) Express \(\displaystyle 7^{2/3}\) in Radical Form The answer is \(\sqrt[3]{7^2}\) or \(\left(\sqrt[3]{7}\right)^2\)
\(\,\,\,\,\,\,7^{2/3}\)
\(\,\,\,\,\,\,\sqrt[3]{7^2}\)
\(\,\,\,\,\,\,\left(\sqrt[3]{7}\right)^2\)
The answer is \(\sqrt[3]{7^2}\) or \(\left(\sqrt[3]{7}\right)^2\)
\(\,\,\,\,\,\,7^{2/3}\)
\(\,\,\,\,\,\,\sqrt[3]{7^2}\)
\(\,\,\,\,\,\,\left(\sqrt[3]{7}\right)^2\)
The answer is \(\sqrt[3]{7^2}\) or \(\left(\sqrt[3]{7}\right)^2\)
\(\textbf{16)}\) Express \(\displaystyle x^{5/6}\) in Radical Form The answer is \(\sqrt[6]{x^5}\) or \(\left(\sqrt[6]{x}\right)^5\)
\(\,\,\,\,\,\,x^{5/6}\)
\(\,\,\,\,\,\,\sqrt[6]{x^5}\)
\(\,\,\,\,\,\,\left(\sqrt[6]{x}\right)^5\)
The answer is \(\sqrt[6]{x^5}\) or \(\left(\sqrt[6]{x}\right)^5\)
\(\,\,\,\,\,\,x^{5/6}\)
\(\,\,\,\,\,\,\sqrt[6]{x^5}\)
\(\,\,\,\,\,\,\left(\sqrt[6]{x}\right)^5\)
The answer is \(\sqrt[6]{x^5}\) or \(\left(\sqrt[6]{x}\right)^5\)
\(\textbf{17)}\) Express \(\displaystyle 10^{1/3}\) in Radical Form The answer is \(\sqrt[3]{10}\)
\(\,\,\,\,\,\,10^{1/3}\)
\(\,\,\,\,\,\,\sqrt[3]{10^1}\)
\(\,\,\,\,\,\,\sqrt[3]{10}\)
The answer is \(\sqrt[3]{10}\)
\(\,\,\,\,\,\,10^{1/3}\)
\(\,\,\,\,\,\,\sqrt[3]{10^1}\)
\(\,\,\,\,\,\,\sqrt[3]{10}\)
The answer is \(\sqrt[3]{10}\)
\(\textbf{18)}\) Express \(\displaystyle a^{7/4}\) in Radical Form The answer is \(\sqrt[4]{a^7}\) or \(\left(\sqrt[4]{a}\right)^7\)
\(\,\,\,\,\,\,a^{7/4}\)
\(\,\,\,\,\,\,\sqrt[4]{a^7}\)
\(\,\,\,\,\,\,\left(\sqrt[4]{a}\right)^7\)
The answer is \(\sqrt[4]{a^7}\) or \(\left(\sqrt[4]{a}\right)^7\)
\(\,\,\,\,\,\,a^{7/4}\)
\(\,\,\,\,\,\,\sqrt[4]{a^7}\)
\(\,\,\,\,\,\,\left(\sqrt[4]{a}\right)^7\)
The answer is \(\sqrt[4]{a^7}\) or \(\left(\sqrt[4]{a}\right)^7\)
\(\textbf{19)}\) Express \(\sqrt[3]{x^2y^5}\) in Rational Exponent Form The answer is \(\displaystyle x^{2/3}y^{5/3}\)
\(\,\,\,\,\,\,\sqrt[3]{x^2y^5}\)
\(\,\,\,\,\,\,\left(x^2y^5\right)^{1/3}\)
\(\,\,\,\,\,\,x^{2/3}y^{5/3}\)
The answer is \(\displaystyle x^{2/3}y^{5/3}\)
\(\,\,\,\,\,\,\sqrt[3]{x^2y^5}\)
\(\,\,\,\,\,\,\left(x^2y^5\right)^{1/3}\)
\(\,\,\,\,\,\,x^{2/3}y^{5/3}\)
The answer is \(\displaystyle x^{2/3}y^{5/3}\)
\(\textbf{20)}\) Express \(\displaystyle a^{3/2}b^{2/5}\) in Radical Form The answer is \(\sqrt{a^3}\sqrt[5]{b^2}\)
\(\,\,\,\,\,\,a^{3/2}b^{2/5}\)
\(\,\,\,\,\,\,a^{3/2}\cdot b^{2/5}\)
\(\,\,\,\,\,\,\sqrt{a^3}\cdot\sqrt[5]{b^2}\)
\(\,\,\,\,\,\,\sqrt{a^3}\sqrt[5]{b^2}\)
The answer is \(\sqrt{a^3}\sqrt[5]{b^2}\)
\(\,\,\,\,\,\,a^{3/2}b^{2/5}\)
\(\,\,\,\,\,\,a^{3/2}\cdot b^{2/5}\)
\(\,\,\,\,\,\,\sqrt{a^3}\cdot\sqrt[5]{b^2}\)
\(\,\,\,\,\,\,\sqrt{a^3}\sqrt[5]{b^2}\)
The answer is \(\sqrt{a^3}\sqrt[5]{b^2}\)
See Related Pages\(\)
\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,3^2=3 \times 3 =9\)
\(\bullet\text{ Negative Exponents}\)
\(\,\,\,\,\,\,\,\,3^{-2}=\frac{1}{9}\)
\(\bullet\text{ Rational Exponents}\)
\(\,\,\,\,\,\,\,\,\displaystyle x^{a/b}=\sqrt[b]{x^a}\)