ACT · Intermediate Algebra / Function
Convert radicals to rational exponents, and convert rational exponents back to radicals.
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Root to denominator
The index of the radical becomes the denominator.
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
\(\sqrt[3]{x^2}\)
The root is 3. The power is 2.
\(x^{\frac{2}{3}}\)
Denominator to root
The denominator of the exponent becomes the index of the radical.
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
\(x^{\frac{1}{2}}\) = \(\sqrt{x}\)
Pull out whole powers
If the numerator is larger than the denominator, split off a whole power so the leftover exponent is less than 1.
\(\frac{3}{2} = 1 + \frac{1}{2}\)
\(x^{\frac{3}{2}} = x \cdot x^{\frac{1}{2}} = x\sqrt{x}\)
More than one variable
Convert each variable the same way. Keep the factors multiplied.
\(\sqrt[3]{x^2 y}\) = \(x^{\frac{2}{3}} y^{\frac{1}{3}}\)
Different denominators
If the denominators are not the same, use a common denominator so there is one root.
\(x^{\frac{1}{2}} y^{\frac{1}{3}}\)
The denominators are 2 and 3. The common denominator is 6.
\(x^{\frac{1}{2}} = x^{\frac{3}{6}}\)
\(y^{\frac{1}{3}} = y^{\frac{2}{6}}\)
\(x^{\frac{3}{6}} y^{\frac{2}{6}} = \sqrt[6]{x^{3} y^{2}}\)
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In Radical to Rational:
The root is the denominator. The power inside is the numerator.
\(\sqrt[5]{x^{2}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 5, so the denominator is 5.
The power inside is the numerator.
x has power 2 inside.
\(x^{\frac{2}{5}}\)
Answer: \(x^{\frac{2}{5}}\)
One variable. Name the root and the power.
\(\sqrt{x}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
x has power 1 inside.
\(x^{\frac{1}{2}}\)
Answer: \(x^{\frac{1}{2}}\)
\(\sqrt[3]{y}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
y has power 1 inside.
\(y^{\frac{1}{3}}\)
Answer: \(y^{\frac{1}{3}}\)
\(\sqrt{x^{3}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
x has power 3 inside.
\(x^{\frac{3}{2}}\)
Answer: \(x^{\frac{3}{2}}\)
\(\sqrt[3]{a^{2}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
a has power 2 inside.
\(a^{\frac{2}{3}}\)
Answer: \(a^{\frac{2}{3}}\)
\(\sqrt[4]{w}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 4, so the denominator is 4.
The power inside is the numerator.
w has power 1 inside.
\(w^{\frac{1}{4}}\)
Answer: \(w^{\frac{1}{4}}\)
Two variables. Convert each factor.
\(\sqrt{x y}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
x has power 1 inside.
\(x^{\frac{1}{2}}\)
y has power 1 inside.
\(y^{\frac{1}{2}}\)
\(x^{\frac{1}{2}} y^{\frac{1}{2}}\)
Answer: \(x^{\frac{1}{2}} y^{\frac{1}{2}}\)
\(\sqrt[3]{a^{2} b}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
a has power 2 inside.
\(a^{\frac{2}{3}}\)
b has power 1 inside.
\(b^{\frac{1}{3}}\)
\(a^{\frac{2}{3}} b^{\frac{1}{3}}\)
Answer: \(a^{\frac{2}{3}} b^{\frac{1}{3}}\)
\(\sqrt{x^{3} y}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
x has power 3 inside.
\(x^{\frac{3}{2}}\)
y has power 1 inside.
\(y^{\frac{1}{2}}\)
\(x^{\frac{3}{2}} y^{\frac{1}{2}}\)
Answer: \(x^{\frac{3}{2}} y^{\frac{1}{2}}\)
\(\sqrt[3]{x y^{2}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
x has power 1 inside.
\(x^{\frac{1}{3}}\)
y has power 2 inside.
\(y^{\frac{2}{3}}\)
\(x^{\frac{1}{3}} y^{\frac{2}{3}}\)
Answer: \(x^{\frac{1}{3}} y^{\frac{2}{3}}\)
\(\sqrt[4]{x^{3} y}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 4, so the denominator is 4.
The power inside is the numerator.
x has power 3 inside.
\(x^{\frac{3}{4}}\)
y has power 1 inside.
\(y^{\frac{1}{4}}\)
\(x^{\frac{3}{4}} y^{\frac{1}{4}}\)
Answer: \(x^{\frac{3}{4}} y^{\frac{1}{4}}\)
Three variables. Reduce an exponent if it is not already in lowest terms.
\(\sqrt[3]{x^{2} y z^{4}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
x has power 2 inside.
\(x^{\frac{2}{3}}\)
y has power 1 inside.
\(y^{\frac{1}{3}}\)
z has power 4 inside.
\(z^{\frac{4}{3}}\)
\(x^{\frac{2}{3}} y^{\frac{1}{3}} z^{\frac{4}{3}}\)
Answer: \(x^{\frac{2}{3}} y^{\frac{1}{3}} z^{\frac{4}{3}}\)
\(\sqrt{a^{5} b^{3} c}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
a has power 5 inside.
\(a^{\frac{5}{2}}\)
b has power 3 inside.
\(b^{\frac{3}{2}}\)
c has power 1 inside.
\(c^{\frac{1}{2}}\)
\(a^{\frac{5}{2}} b^{\frac{3}{2}} c^{\frac{1}{2}}\)
Answer: \(a^{\frac{5}{2}} b^{\frac{3}{2}} c^{\frac{1}{2}}\)
\(\sqrt[3]{x^{4} y^{2} z}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
x has power 4 inside.
\(x^{\frac{4}{3}}\)
y has power 2 inside.
\(y^{\frac{2}{3}}\)
z has power 1 inside.
\(z^{\frac{1}{3}}\)
\(x^{\frac{4}{3}} y^{\frac{2}{3}} z^{\frac{1}{3}}\)
Answer: \(x^{\frac{4}{3}} y^{\frac{2}{3}} z^{\frac{1}{3}}\)
\(\sqrt[5]{x^{2} y^{3} z}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 5, so the denominator is 5.
The power inside is the numerator.
x has power 2 inside.
\(x^{\frac{2}{5}}\)
y has power 3 inside.
\(y^{\frac{3}{5}}\)
z has power 1 inside.
\(z^{\frac{1}{5}}\)
\(x^{\frac{2}{5}} y^{\frac{3}{5}} z^{\frac{1}{5}}\)
Answer: \(x^{\frac{2}{5}} y^{\frac{3}{5}} z^{\frac{1}{5}}\)
\(\sqrt[4]{x^{3} y z^{2}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 4, so the denominator is 4.
The power inside is the numerator.
x has power 3 inside.
\(x^{\frac{3}{4}}\)
y has power 1 inside.
\(y^{\frac{1}{4}}\)
z has power 2 inside.
\(z^{\frac{2}{4}}\)
\(z^{\frac{1}{2}}\)
\(x^{\frac{3}{4}} y^{\frac{1}{4}} z^{\frac{1}{2}}\)
Answer: \(x^{\frac{3}{4}} y^{\frac{1}{4}} z^{\frac{1}{2}}\)
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In Rational to Radical:
The denominator is the root. The numerator is the power. Pull out whole powers so each leftover exponent is less than 1. On hard problems, use a common denominator first so there is one root.
\(y^{\frac{5}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
If the numerator is larger than the denominator, split off a whole power first.
\(\frac{5}{2} = 2 + \frac{1}{2}\)
\(y^{\frac{5}{2}} = y^{2} \cdot y^{\frac{1}{2}}\)
\(y^{\frac{1}{2}} = \sqrt{y}\)
\(y^{2} \sqrt{y}\)
Answer: \(y^{2} \sqrt{y}\)
One variable. Write the root, and pull out a whole power when the numerator is larger.
\(x^{\frac{1}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(x^{\frac{1}{2}}\) has root 2 and power 1.
\(x^{\frac{1}{2}} = \sqrt{x}\)
Answer: \(\sqrt{x}\)
\(y^{\frac{1}{3}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(y^{\frac{1}{3}}\) has root 3 and power 1.
\(y^{\frac{1}{3}} = \sqrt[3]{y}\)
Answer: \(\sqrt[3]{y}\)
\(a^{\frac{2}{3}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(a^{\frac{2}{3}}\) has root 3 and power 2.
\(a^{\frac{2}{3}} = \sqrt[3]{a^{2}}\)
Answer: \(\sqrt[3]{a^{2}}\)
\(z^{\frac{3}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
If the numerator is larger than the denominator, split off a whole power first.
\(\frac{3}{2} = 1 + \frac{1}{2}\)
\(z^{\frac{3}{2}} = z \cdot z^{\frac{1}{2}}\)
\(z^{\frac{1}{2}} = \sqrt{z}\)
\(z \sqrt{z}\)
Answer: \(z \sqrt{z}\)
\(b^{\frac{1}{4}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(b^{\frac{1}{4}}\) has root 4 and power 1.
\(b^{\frac{1}{4}} = \sqrt[4]{b}\)
Answer: \(\sqrt[4]{b}\)
Two variables with the same root. Combine what is left into one simplified radical.
\(x^{\frac{1}{2}} y^{\frac{1}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(x^{\frac{1}{2}}\) has root 2 and power 1.
\(y^{\frac{1}{2}}\) has root 2 and power 1.
Same root, so write one radical.
\(\sqrt{x y}\)
\(\sqrt{x y}\)
Answer: \(\sqrt{x y}\)
\(a^{\frac{2}{3}} b^{\frac{1}{3}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(a^{\frac{2}{3}}\) has root 3 and power 2.
\(b^{\frac{1}{3}}\) has root 3 and power 1.
Same root, so write one radical.
\(\sqrt[3]{a^{2} b}\)
\(\sqrt[3]{a^{2} b}\)
Answer: \(\sqrt[3]{a^{2} b}\)
\(x^{\frac{3}{2}} y^{\frac{1}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
If the numerator is larger than the denominator, split off a whole power first.
\(\frac{3}{2} = 1 + \frac{1}{2}\)
\(x^{\frac{3}{2}} = x \cdot x^{\frac{1}{2}}\)
\(y^{\frac{1}{2}}\) has root 2 and power 1.
Same root on what is left. Write one radical.
\(\sqrt{x y}\)
\(x \sqrt{x y}\)
Answer: \(x \sqrt{x y}\)
\(x^{\frac{1}{3}} y^{\frac{2}{3}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(x^{\frac{1}{3}}\) has root 3 and power 1.
\(y^{\frac{2}{3}}\) has root 3 and power 2.
Same root, so write one radical.
\(\sqrt[3]{x y^{2}}\)
\(\sqrt[3]{x y^{2}}\)
Answer: \(\sqrt[3]{x y^{2}}\)
\(x^{\frac{3}{4}} y^{\frac{1}{4}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
\(x^{\frac{3}{4}}\) has root 4 and power 3.
\(y^{\frac{1}{4}}\) has root 4 and power 1.
Same root, so write one radical.
\(\sqrt[4]{x^{3} y}\)
\(\sqrt[4]{x^{3} y}\)
Answer: \(\sqrt[4]{x^{3} y}\)
Different denominators. Use a common denominator, pull out whole powers, then write one radical.
\(x^{\frac{5}{2}} y^{\frac{1}{3}}\)
The denominators are 2 and 3. The common denominator is 6.
\(\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6}\)
\(x^{\frac{5}{2}} = x^{\frac{15}{6}}\)
\(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)
\(y^{\frac{1}{3}} = y^{\frac{2}{6}}\)
\(x^{\frac{15}{6}} y^{\frac{2}{6}}\)
\(\frac{15}{6} = 2 + \frac{3}{6}\)
\(x^{\frac{15}{6}} = x^{2} \cdot x^{\frac{3}{6}}\)
\(y^{\frac{2}{6}}\) is less than 1, so it stays inside the root.
Same root on what is left. Write one radical.
\(x^{\frac{3}{6}} y^{\frac{2}{6}} = \sqrt[6]{x^{3} y^{2}}\)
\(x^{2} \sqrt[6]{x^{3} y^{2}}\)
Answer: \(x^{2} \sqrt[6]{x^{3} y^{2}}\)
\(a^{\frac{7}{3}} b^{\frac{1}{2}}\)
The denominators are 3 and 2. The common denominator is 6.
\(\frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}\)
\(a^{\frac{7}{3}} = a^{\frac{14}{6}}\)
\(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\)
\(b^{\frac{1}{2}} = b^{\frac{3}{6}}\)
\(a^{\frac{14}{6}} b^{\frac{3}{6}}\)
\(\frac{14}{6} = 2 + \frac{2}{6}\)
\(a^{\frac{14}{6}} = a^{2} \cdot a^{\frac{2}{6}}\)
\(b^{\frac{3}{6}}\) is less than 1, so it stays inside the root.
Same root on what is left. Write one radical.
\(a^{\frac{2}{6}} b^{\frac{3}{6}} = \sqrt[6]{a^{2} b^{3}}\)
\(a^{2} \sqrt[6]{a^{2} b^{3}}\)
Answer: \(a^{2} \sqrt[6]{a^{2} b^{3}}\)
\(x^{\frac{5}{4}} y^{\frac{2}{3}}\)
The denominators are 4 and 3. The common denominator is 12.
\(\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}\)
\(x^{\frac{5}{4}} = x^{\frac{15}{12}}\)
\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)
\(y^{\frac{2}{3}} = y^{\frac{8}{12}}\)
\(x^{\frac{15}{12}} y^{\frac{8}{12}}\)
\(\frac{15}{12} = 1 + \frac{3}{12}\)
\(x^{\frac{15}{12}} = x \cdot x^{\frac{3}{12}}\)
\(y^{\frac{8}{12}}\) is less than 1, so it stays inside the root.
Same root on what is left. Write one radical.
\(x^{\frac{3}{12}} y^{\frac{8}{12}} = \sqrt[12]{x^{3} y^{8}}\)
\(x \sqrt[12]{x^{3} y^{8}}\)
Answer: \(x \sqrt[12]{x^{3} y^{8}}\)
\(m^{\frac{1}{3}} n^{\frac{1}{4}}\)
The denominators are 3 and 4. The common denominator is 12.
\(\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)
\(m^{\frac{1}{3}} = m^{\frac{4}{12}}\)
\(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)
\(n^{\frac{1}{4}} = n^{\frac{3}{12}}\)
\(m^{\frac{4}{12}} n^{\frac{3}{12}}\)
\(m^{\frac{4}{12}}\) is less than 1, so it stays inside the root.
\(n^{\frac{3}{12}}\) is less than 1, so it stays inside the root.
Same root on what is left. Write one radical.
\(m^{\frac{4}{12}} n^{\frac{3}{12}} = \sqrt[12]{m^{4} n^{3}}\)
\(\sqrt[12]{m^{4} n^{3}}\)
Answer: \(\sqrt[12]{m^{4} n^{3}}\)
\(x^{\frac{7}{2}} y^{\frac{4}{3}} z^{\frac{1}{5}}\)
The denominators are 2, 3, and 5. The common denominator is 30.
\(\frac{7}{2} = \frac{7 \times 15}{2 \times 15} = \frac{105}{30}\)
\(x^{\frac{7}{2}} = x^{\frac{105}{30}}\)
\(\frac{4}{3} = \frac{4 \times 10}{3 \times 10} = \frac{40}{30}\)
\(y^{\frac{4}{3}} = y^{\frac{40}{30}}\)
\(\frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30}\)
\(z^{\frac{1}{5}} = z^{\frac{6}{30}}\)
\(x^{\frac{105}{30}} y^{\frac{40}{30}} z^{\frac{6}{30}}\)
\(\frac{105}{30} = 3 + \frac{15}{30}\)
\(x^{\frac{105}{30}} = x^{3} \cdot x^{\frac{15}{30}}\)
\(\frac{40}{30} = 1 + \frac{10}{30}\)
\(y^{\frac{40}{30}} = y \cdot y^{\frac{10}{30}}\)
\(z^{\frac{6}{30}}\) is less than 1, so it stays inside the root.
Same root on what is left. Write one radical.
\(x^{\frac{15}{30}} y^{\frac{10}{30}} z^{\frac{6}{30}} = \sqrt[30]{x^{15} y^{10} z^{6}}\)
\(x^{3} y \sqrt[30]{x^{15} y^{10} z^{6}}\)
Answer: \(x^{3} y \sqrt[30]{x^{15} y^{10} z^{6}}\)
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Six multiple-choice questions. They mix both conversions and all three levels. Show Answer names the section and level.
Radical to Rational · Easy
\(\sqrt[3]{m^{4}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
m has power 4 inside.
\(m^{\frac{4}{3}}\)
Answer: C
Rational to Radical · Easy
\(k^{\frac{5}{2}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
If the numerator is larger than the denominator, split off a whole power first.
\(\frac{5}{2} = 2 + \frac{1}{2}\)
\(k^{\frac{5}{2}} = k^{2} \cdot k^{\frac{1}{2}}\)
\(k^{\frac{1}{2}} = \sqrt{k}\)
\(k^{2} \sqrt{k}\)
Answer: A
Radical to Rational · Medium
\(\sqrt{p^{5} q}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 2, so the denominator is 2.
The power inside is the numerator.
p has power 5 inside.
\(p^{\frac{5}{2}}\)
q has power 1 inside.
\(q^{\frac{1}{2}}\)
\(p^{\frac{5}{2}} q^{\frac{1}{2}}\)
Answer: D
Rational to Radical · Medium
\(u^{\frac{4}{3}} v^{\frac{2}{3}}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
The denominator is the root. The numerator is the power.
If the numerator is larger than the denominator, split off a whole power first.
\(\frac{4}{3} = 1 + \frac{1}{3}\)
\(u^{\frac{4}{3}} = u \cdot u^{\frac{1}{3}}\)
\(v^{\frac{2}{3}}\) has root 3 and power 2.
Same root on what is left. Write one radical.
\(\sqrt[3]{u v^{2}}\)
\(u \sqrt[3]{u v^{2}}\)
Answer: B
Radical to Rational · Hard
\(\sqrt[3]{p^{5} q r^{2}}\)
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
The root is 3, so the denominator is 3.
The power inside is the numerator.
p has power 5 inside.
\(p^{\frac{5}{3}}\)
q has power 1 inside.
\(q^{\frac{1}{3}}\)
r has power 2 inside.
\(r^{\frac{2}{3}}\)
\(p^{\frac{5}{3}} q^{\frac{1}{3}} r^{\frac{2}{3}}\)
Answer: E
Rational to Radical · Hard
\(a^{\frac{5}{3}} b^{\frac{5}{2}}\)
The denominators are 3 and 2. The common denominator is 6.
\(\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}\)
\(a^{\frac{5}{3}} = a^{\frac{10}{6}}\)
\(\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6}\)
\(b^{\frac{5}{2}} = b^{\frac{15}{6}}\)
\(a^{\frac{10}{6}} b^{\frac{15}{6}}\)
\(\frac{10}{6} = 1 + \frac{4}{6}\)
\(a^{\frac{10}{6}} = a \cdot a^{\frac{4}{6}}\)
\(\frac{15}{6} = 2 + \frac{3}{6}\)
\(b^{\frac{15}{6}} = b^{2} \cdot b^{\frac{3}{6}}\)
Same root on what is left. Write one radical.
\(a^{\frac{4}{6}} b^{\frac{3}{6}} = \sqrt[6]{a^{4} b^{3}}\)
\(a b^{2} \sqrt[6]{a^{4} b^{3}}\)
Answer: A
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