Fractional Exponents & Radicals

Rules and worked solutions · Problems 3, 5–10, and 13–28

Exponent rules

These rules apply to the positive bases used in these problems. Denominators must be nonzero.

1. Fractional exponent = root

a1/n = ⁿ√a
The denominator tells you which root to take: 2 = square root, 3 = cube root, 4 = fourth root.

2. The numerator = power

am/n = (ⁿ√a)m = ⁿ√(am)
Take the root and then raise it to the numerator power. Either order works for positive bases.

3. Negative exponent = reciprocal

a−r = 1/ar
For a fraction, a negative exponent flips it:
(a/b)−r = (b/a)r

4. Power raised to a power

(am)n = amn
Multiply the exponents.

5. Divide powers with the same base (Quotient rule)

am / an = am−n
Keep the base and subtract the exponents. A plain number such as 6 is the same as 61.

6. Matching roots and powers

ⁿ√(an/bn) = a/b
For positive numbers, matching roots and powers cancel.

7. Multiply powers with the same base (Product rule)

am · an = am+n
Keep the base and add the exponents, including their signs.

Add exponents when multiplying powers with the same base. Multiply exponents when raising a power to another power.

Radical rules

The numbers under the radical signs in these problems are positive. The index is the small number that tells you which root to take. A square root has an understood index of 2.

8. Multiply radicals with the same index (Product rule for radicals)

ⁿ√a · ⁿ√b = ⁿ√(ab)
Multiply the numbers inside and keep the same root.
√2 · √8 = √16 = 4

The indices must match before you combine radicals this way.

9. Divide radicals with the same index (Quotient rule for radicals)

ⁿ√a / ⁿ√b = ⁿ√(a/b)
Divide the numbers inside and keep the same root. Simplify the fraction inside when possible.

10. Pull out perfect powers

ⁿ√(cn · d) = c · ⁿ√d
For positive c, a group of n matching factors comes out as one factor.
  • Square roots: look for perfect squares such as 4, 9, 16, and 25.
  • Cube roots: look for perfect cubes such as 8, 27, and 64.
  • Fourth roots: look for perfect fourth powers such as 16 and 81.
  • Fifth roots: look for perfect fifth powers such as 32 and 243.

11. Rationalize the denominator

Multiply the numerator and denominator by the same nonzero radical so the denominator becomes a whole number.
1∛4 · ∛2∛2 = ∛2∛8 = ∛22
Here, 4 needs one more factor of 2 to become the perfect cube 8.

12. Reduce the root index when possible

Rewrite the radical as a fractional exponent and reduce the fraction.
⁴√4 = (22)1/4 = 21/2 = √2
This is useful even when no whole-number factor can be pulled out.

Fractional exponent problems

Problem 3

(92)1/3

Rules used: 4. Power raised to a power; 1–2. Fractional exponent.
Step 1. Multiply the exponents.
(92)1/3 = 92·(1/3) = 92/3
Step 2. Rewrite as a cube root.
92/3 = ∛(92) = ∛81
Step 3. Factor out the perfect cube 27.
∛81 = ∛(27 · 3) = 3∛3
Exact answer
3∛3
Problem 5

6 / 61/4

Rule used: 5. Divide powers with the same base (Quotient rule).
Step 1. Rewrite 6 as 61.
61 / 61/4
Step 2. Subtract the exponents.
61 − 1/4 = 63/4
Step 3. Rewrite in radical form.
63/4 = ⁴√(63) = ⁴√216
Exact answer
⁴√216
Also written as 63/4.
Problem 6

7 / 71/3

Rule used: 5. Divide powers with the same base (Quotient rule).
Step 1. Subtract the exponents.
71 / 71/3 = 71 − 1/3 = 72/3
Step 2. Rewrite in radical form.
72/3 = ∛(72) = ∛49
Exact answer
∛49
Also written as 72/3.
Problem 7

(84/104)−1/4

Rules used: 3. Negative exponent; 6. Matching roots and powers.
Step 1. The negative exponent flips the fraction.
(104/84)1/4
Step 2. Take the fourth roots.
10/8
Step 3. Reduce the fraction.
Exact answer
5/4
Problem 8

(93/63)−1/3

Rules used: 3. Negative exponent; 6. Matching roots and powers.
Step 1. The negative exponent flips the fraction.
(63/93)1/3
Step 2. Take the cube roots.
6/9
Step 3. Reduce the fraction.
Exact answer
2/3
Problem 9

(3−2/3 · 31/3)−1

Rules used: 7. Multiply powers with the same base (Product rule); 4. Power raised to a power.
Step 1. Add the exponents inside the parentheses.
3−2/3 · 31/3 = 3−2/3 + 1/3 = 3−1/3
Step 2. Multiply by the outside exponent.
(3−1/3)−1 = 3(−1/3)(−1) = 31/3
Step 3. Rewrite as a cube root.
Exact answer
∛3
Problem 10

(51/2 · 5−3/2)−1/4

Rules used: 7. Multiply powers with the same base (Product rule); 4. Power raised to a power.
Step 1. Add the exponents inside the parentheses.
51/2 · 5−3/2 = 51/2 − 3/2 = 5−1
Step 2. Multiply by the outside exponent.
(5−1)−1/4 = 5(−1)(−1/4) = 51/4
Step 3. Rewrite as a fourth root.
Exact answer
⁴√5

Problems 13–20: Use properties of radicals

Problem 13

√2 · √72

Rule used: Product rule for radicals.
Step 1. Both radicals are square roots, so multiply inside.
√2 · √72 = √(2 · 72) = √144
Step 2. Since 122 = 144, take the square root.
Exact answer
12
Problem 14

∛16 · ∛32

Rule used: Product rule for radicals.
Step 1. Combine the cube roots.
∛16 · ∛32 = ∛(16 · 32) = ∛512
Step 2. Since 83 = 512, take the cube root.
Exact answer
8
Problem 15

⁴√6 · ⁴√8

Rules used: Product rule; pull out perfect powers.
Step 1. Combine the fourth roots.
⁴√6 · ⁴√8 = ⁴√48
Step 2. Factor 48 using the perfect fourth power 16.
⁴√48 = ⁴√(16 · 3) = ⁴√16 · ⁴√3
Step 3. Since 24 = 16, pull out 2.
Exact answer
2⁴√3
This means 2 times the fourth root of 3.
Problem 16

⁴√8 · ⁴√8

Rules used: Product rule; perfect powers; reduce the root index.
Step 1. Combine the fourth roots.
⁴√8 · ⁴√8 = ⁴√64
Step 2. Factor out the perfect fourth power 16.
⁴√64 = ⁴√(16 · 4) = 2 · ⁴√4
Step 3. Reduce the remaining fourth root to a square root.
⁴√4 = (22)1/4 = 21/2 = √2
Exact answer
2√2
Problem 17

⁵√486⁵√2

Rule used: Quotient rule for radicals.
Step 1. Combine the fifth roots and divide inside.
⁵√486 / ⁵√2 = ⁵√(486/2) = ⁵√243
Step 2. Since 35 = 243, take the fifth root.
Exact answer
3
Problem 18

√2√32

Rule used: Quotient rule for radicals.
Step 1. Combine the square roots and simplify the fraction inside.
√2 / √32 = √(2/32) = √(1/16)
Step 2. Take the square root of the numerator and denominator.
√(1/16) = √1 / √16 = 1/4
Exact answer
1/4
Problem 19

∛6 · ∛72 ∛2

Rules used: Product and quotient rules for radicals.
Step 1. All three radicals are cube roots, so combine them.
∛6 · ∛72 ∛2 = ∛((6 · 72)/2)
Step 2. Simplify inside the cube root.
∛((6 · 72)/2) = ∛216
Step 3. Since 63 = 216, take the cube root.
Exact answer
6
Problem 20

∛3 · ∛18 ⁶√2 · ⁶√2

Rules used: Product rule; fractional exponents; quotient rule.
Step 1. Combine the cube roots in the numerator.
∛3 · ∛18 = ∛54
Step 2. Rewrite the denominator using exponents. Add the exponents because the bases match.
⁶√2 · ⁶√2 = 21/6 · 21/6 = 22/6 = 21/3 = ∛2
Step 3. Now both roots have index 3, so divide inside.
∛54 / ∛2 = ∛(54/2) = ∛27 = 3
Exact answer
3
Do not combine a cube root and a sixth root directly using the quotient rule. First rewrite them with matching indices.

Problems 21–28: Write in simplest form

Problem 21

⁴√567

Rule used: Pull out perfect powers.
Step 1. Factor 567 using the perfect fourth power 81.
567 = 81 · 7 = 34 · 7
Step 2. Take the fourth root of 81.
⁴√567 = ⁴√(81 · 7) = 3 · ⁴√7
Exact answer
3⁴√7
This means 3 times the fourth root of 7.
Problem 22

⁵√288

Rule used: Pull out perfect powers.
Step 1. Factor 288 using the perfect fifth power 32.
288 = 32 · 9 = 25 · 9
Step 2. Take the fifth root of 32.
⁵√288 = ⁵√(32 · 9) = 2 · ⁵√9
Exact answer
2⁵√9
This means 2 times the fifth root of 9. The remaining 9 has no perfect fifth-power factor greater than 1.
Problem 23

∛5∛4

Rule used: Rationalize the denominator.
Step 1. The denominator is ∛4 = ∛(22). It needs one more factor of 2 inside to make the perfect cube 8. Multiply the numerator and denominator by ∛2.
∛5∛4 · ∛2∛2 = ∛10∛8
Step 2. Simplify ∛8 to 2.
Exact answer
∛102
Problem 24

⁴√4⁴√27

Rule used: Rationalize the denominator.
Step 1. Since 27 = 33, multiply by ⁴√3 / ⁴√3 to make the denominator the fourth root of 34.
⁴√4⁴√27 · ⁴√3⁴√3 = ⁴√12⁴√81
Step 2. Since 34 = 81, the denominator becomes 3.
Exact answer
⁴√123
Problem 25

√(3/8)

Rules used: Quotient rule; simplify radicals; rationalize the denominator.
Step 1. Separate the square roots and simplify √8.
√(3/8) = √3√8 = √32√2
Step 2. Multiply the numerator and denominator by √2.
√32√2 · √2√2 = √62 · 2
Exact answer
√64
Problem 26

∛(7/4)

Rules used: Quotient rule; rationalize the denominator.
Step 1. Separate the cube roots.
∛(7/4) = ∛7∛4
Step 2. Multiply the numerator and denominator by ∛2. This makes the denominator ∛8.
∛7∛4 · ∛2∛2 = ∛14∛8
Step 3. Simplify ∛8 to 2.
Exact answer
∛142
Problem 27

∛(64/49)

Rules used: Quotient rule; perfect powers; rationalize the denominator.
Step 1. Separate the cube roots and simplify ∛64.
∛(64/49) = ∛64∛49 = 4∛49
Step 2. Since 49 = 72, multiply by ∛7 / ∛7 to make the denominator the cube root of 73.
4∛49 · ∛7∛7 = 4∛7∛343
Step 3. Simplify ∛343 to 7.
Exact answer
4∛77
Problem 28

⁴√(1296/25)

Rules used: Quotient rule; perfect powers; reduce the root index; rationalize the denominator.
Step 1. Separate the fourth roots. Since 64 = 1296, the numerator becomes 6.
⁴√(1296/25) = ⁴√1296⁴√25 = 6⁴√25
Step 2. Rewrite the denominator as a square root.
⁴√25 = (52)1/4 = 51/2 = √5
Step 3. Multiply the numerator and denominator by √5.
6√5 · √5√5 = 6√55
Exact answer
6√55

Quick checklist

  1. Multiplying powers with the same base? Add the exponents.
  2. Dividing powers with the same base? Subtract the exponents.
  3. Raising a power to a power? Multiply the exponents.
  4. Negative exponent? Use the reciprocal, or combine exponents first when helpful.
  5. Fractional exponent? Denominator = root; numerator = power.
  6. Multiplying or dividing radicals? Make sure the root indices match before combining them.
  7. Perfect-power factors inside a radical? Pull them out using the root index.
  8. Can the root index be reduced? Rewrite using fractional exponents and simplify.
  9. Radical in the denominator? Rationalize by multiplying the numerator and denominator by the same suitable radical.
  10. Final check: Reduce fractions and make sure no further radical simplification is possible.

Tip: Simplify before calculating large powers. When rationalizing a denominator, look for the missing factors that will make the number inside a perfect square, cube, fourth power, or fifth power.