Fractional Indices: Rules, Formulas and Examples

Fractional Indices represent powers written as fractions, such as a^(m/n). The denominator indicates a root, while the numerator indicates a power. This page explains the fractional exponent formula, laws of fractional indices, negative fractional powers, domain restrictions and quick methods for solving numerical questions accurately.

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What Are Fractional Indices?

A fractional index is an exponent written as a fraction: a^(m/n), where m and n are integers and n ≠ 0. For a positive base, a^(m/n) means the nth root of a raised to the power m.

The basic conversion is a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m). Thus, the denominator gives the root and the numerator gives the power. For example, 16^(3/4) = (⁴√16)^3 = 2^3 = 8. The fraction should generally be reduced before evaluation, because the reduced denominator determines the simplest root form.

Fractional Indices Formula & Tricks

Important Formulas

Fractional exponent formula
a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m

The denominator n represents the root and the numerator m represents the power.

Negative fractional exponent
a^(-m/n) = 1/a^(m/n) = 1/(ⁿ√(a^m))

A negative fractional index takes the reciprocal of the corresponding positive fractional power.

Product rule
a^p × a^q = a^(p+q)

For the same non-zero base, add the indices.

Quotient rule
a^p ÷ a^q = a^(p−q)

For the same non-zero base, subtract the denominator index from the numerator index.

Power of a power
(a^p)^q = a^(pq)

Multiply the indices when a power is raised to another power.

Quick Tricks

Reduce the index before taking the root

Reduce m/n to its lowest terms, then use the denominator as the root. This often changes a complicated-looking expression into a simple perfect-root calculation.

Example: 125^(6/9) = 125^(2/3) = (³√125)^2 = 5^2 = 25.
Take the root before the power when convenient

For perfect powers, calculate the root first and then apply the numerator power. This usually keeps the numbers smaller.

Example: 256^(3/4) = (⁴√256)^3 = 4^3 = 64.
Handle a negative index last

Evaluate the positive fractional power first and then take its reciprocal.

Example: 16^(-3/4) = 1/16^(3/4) = 1/8.

Fractional Indices Concepts

Converting Fractional Indices into Roots

In a^(m/n), n is the root order and m is the power applied to the root.

Use a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m. Both forms have the same value when the expression is defined. For example, 32^(2/5) = (⁵√32)^2 = 2^2 = 4.

Example: 81^(3/4) = (⁴√81)^3 = 3^3 = 27.

Evaluating Fractional Powers of Perfect Powers

To evaluate a fractional power, identify the required root and then apply the numerator power.

If the base is a perfect nth power, the calculation becomes direct. Prime factorisation can also be used: a^(m/n) is evaluated by dividing the prime-factor exponents by n and multiplying them by m.

Example: 64^(5/6) = (⁶√64)^5 = 2^5 = 32.

Negative Fractional Indices

A negative fractional index gives the reciprocal of the corresponding positive fractional power.

For a ≠ 0, a^(-m/n) = 1/a^(m/n). The negative sign applies to the complete fractional exponent, not only to its numerator or denominator.

Example: 27^(-2/3) = 1/(27^(2/3)) = 1/(³√27)^2 = 1/9.

Laws of Fractional Indices

The ordinary laws of indices apply to fractional indices when the expressions are defined and the bases meet the required conditions.

For the same positive base, add indices during multiplication, subtract them during division, and multiply them when raising a power to another power. For example, 8^(1/3) × 8^(2/3) = 8^1 = 8.

Example: 16^(3/4) ÷ 16^(1/4) = 16^(2/4) = 16^(1/2) = 4.

Sign and Domain Conditions

For real-number calculations, a positive base is always safe, while a negative base depends on the denominator of the reduced fractional index.

If the reduced denominator is odd, a negative base can have a real value; if it is even, the expression has no real value. For example, (−8)^(1/3) = −2, but (−8)^(1/2) is not real. Index laws involving roots should be applied only when all expressions are defined.

Example: (−32)^(2/5) = ((⁵√(−32)))^2 = (−2)^2 = 4.

Fractional Indices Video Lessons

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Indices: Power, Negative and Fractional Exponents

Understand power of a power, negative exponents, and fractional exponents in indices, including the rules used to simplify expressions accurately.

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Practice Fractional Indices Questions

Practise published questions related to this topic.

1The value of (27^(2/3) × 9^(-1/2))/(3⁻¹ × 81^(1/4)) is→ 2Simplify ((32)6)5 and express it as a power of 9. What is the exponent of 9?→ 3In the records, the value is noted as 7/√7 .Simplify.→ 4If x2 × x3 = xa and x²/(x^b) = x1, what is the value of a + b?→ 5If Y^x = 121 , what will (Y - 1)^((x - 2)) = ?→ 6Simplify: √(144 × 49) + √(64 × 25) - √(100 × 36)→ 7If 6x - 6x-1 = 180, then the value of (2x-1)/(2x+3) is:→ 8Simplify: ((2x²y⁻¹)³)/((4xy)⁴) × ((4x⁻³y⁴)³)/((2x²y)³)→ 9Choose the incorrect relation from the following: I. √6 + √2 = √5 + √3 II. √6 + √2 √5 + √3→ 10The simplified value of the expression 0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a is 1, then the value of a is ______.→

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Quick Revision Notes

Fractional Indices Revision Points

Use these rules to convert, simplify and evaluate fractional powers.

  • a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m.
  • The denominator of the index gives the root; the numerator gives the power.
  • Reduce the fractional index before evaluating the expression.
  • a^(−m/n) = 1/a^(m/n), provided a ≠ 0.
  • For the same positive base, a^p × a^q = a^(p+q) and a^p ÷ a^q = a^(p−q).
  • (a^p)^q = a^(pq).
  • A negative base has a real value when the reduced denominator is odd; an even reduced denominator gives no real value.

Fractional Indices FAQs

What is the fractional exponent formula?

The formula is a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m. The denominator n gives the root and the numerator m gives the power.

How do you calculate 256^(3/4)?

Take the fourth root first: 256^(3/4) = (⁴√256)^3 = 4^3 = 64.

How do you simplify 125^(6/9)?

Reduce 6/9 to 2/3. Therefore, 125^(6/9) = 125^(2/3) = (³√125)^2 = 25.

What is the value of 16^(-3/4)?

16^(-3/4) = 1/16^(3/4) = 1/(⁴√16)^3 = 1/8.

Can a negative number have a fractional index?

Yes, in real numbers when the reduced denominator of the index is odd. For example, (−27)^(2/3) = (−3)^2 = 9, whereas (−27)^(1/2) is not real.

What is the value of 8^(1/3) × 8^(2/3)?

Add the indices: 8^(1/3 + 2/3) = 8^1 = 8.

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