1.5.1 Laws of indices

What you'll learn

How to apply the laws of indices when multiplying, dividing and raising powers to powers.
What zero, negative and fractional indices mean.
How to work with any rational exponent and state necessary restrictions.
How to simplify expressions and solve equations involving indices.

Starting point: powers and indices

A power is an expression such as $a^n$ . The number or expression $a$ is the base, and $n$ is the index or exponent.

For a positive integer $n$ ,

factors.a^n=\underbrace{a\cdot a\cdot \ldots \cdot a}_{n\text{ factors}}.

For example, $x4=x⋅x⋅x⋅xx^4=x\cdot x\cdot x\cdot x$ .

Definition

Rational exponent

A rational exponent is an exponent that can be written as a fraction $mn\frac{m}{n}$ , where $m$ and $n$ are integers and $n≠0n\neq 0$ . Integers are also rational because, for example, $3=313=\frac{3}{1}$ .

Multiplying powers with the same base

When powers with the same base are multiplied, add their indices:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

This follows from counting the total number of factors of $a$ . For example, $a3⋅a2a^3\cdot a^2$ contains three factors of $a$ followed by another two, giving five factors altogether.

Key Idea

Product law

Keep the base unchanged and add the indices:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

Example

Multiplying powers

Simplify $3x5/2⋅4x7/33x^{5/2}\cdot 4x^{7/3}$ .

Multiply the numerical coefficients: $3⋅4=123\cdot 4=12$ .
Add the indices of the common base $x$ :
$52+73=156+146=296.\frac{5}{2}+\frac{7}{3} =\frac{15}{6}+\frac{14}{6} =\frac{29}{6}.$
Therefore,
$3x5/2⋅4x7/3=12x29/6.3x^{5/2}\cdot 4x^{7/3}=12x^{29/6}.$

Common Mistake

Multiplying the indices

Do not use $am⋅an=amna^m\cdot a^n=a^{mn}$ . The indices are added, not multiplied. Multiplication of indices belongs to the power-of-a-power law.

Dividing powers with the same base

When powers with the same non-zero base are divided, subtract the index in the denominator from the index in the numerator:

aman=am−n,a≠0.\frac{a^m}{a^n}=a^{m-n},\qquad a\neq 0.

The restriction $a≠0a\neq 0$ is needed because division by zero is undefined.

Key Idea

Quotient law

Keep the base unchanged and subtract the indices in the order top minus bottom:

aman=am−n.\frac{a^m}{a^n}=a^{m-n}.

Example

Dividing powers

Simplify

18p7/4q56p1/2q8,\frac{18p^{7/4}q^5}{6p^{1/2}q^8},

where $p≠0p\neq 0$ and $q≠0q\neq 0$ .

Divide the coefficients: $18÷6=318\div 6=3$ .
Subtract the indices of $p$ :
$74−12=74−24=54.\frac{7}{4}-\frac{1}{2} =\frac{7}{4}-\frac{2}{4} =\frac{5}{4}.$
Subtract the indices of $q$ : $5 - 8 = - 3$ .
Hence,
$18p7/4q56p1/2q8=3p5/4q−3=3p5/4q3.\frac{18p^{7/4}q^5}{6p^{1/2}q^8} =3p^{5/4}q^{-3} =\frac{3p^{5/4}}{q^3}.$

Raising a power to another power

When one power is raised to another power, multiply the indices:

(am)n=amn.\left(a^m\right)^n=a^{mn}.

Brackets matter: the outer index applies to the whole expression inside them.

Example

Raising a power to a power

Simplify $(8x6)2/3\left(8x^6\right)^{2/3}$ , assuming $x≥0x\geq 0$ .

Apply the outer index to both factors:
$(8x6)2/3=82/3(x6)2/3.\left(8x^6\right)^{2/3}=8^{2/3}\left(x^6\right)^{2/3}.$
Evaluate the numerical power using the cube root:
$82/3=(83)2=22=4.8^{2/3}=\left(\sqrt[3]{8}\right)^2=2^2=4.$
Multiply the indices on $x$ :
$(x6)2/3=x6⋅2/3=x4.\left(x^6\right)^{2/3}=x^{6\cdot 2/3}=x^4.$
Therefore,
$(8x6)2/3=4x4.\left(8x^6\right)^{2/3}=4x^4.$

Powers of products and quotients

An index can be distributed across multiplication:

ab)^n=a^nb^n.

It can also be distributed across a quotient, provided the denominator is non-zero:

(ab)n=anbn,b≠0.\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\qquad b\neq 0.

Common Mistake

Powers do not distribute over addition

In general, $(a+b)n≠an+bn(a+b)^n\neq a^n+b^n$ . For example, $2+3)^2=25$ , but $2^2+3^2=13$ . The distribution laws apply to products and quotients, not sums or differences.

The zero index

For any non-zero base,

a^0=1.

This follows from the quotient law:

amam=am−m=a0.\frac{a^m}{a^m}=a^{m-m}=a^0.

But the fraction on the left equals 1, so $a^0=1$ .

Common Mistake

The base cannot be zero

The zero-index law requires $a≠0a\neq 0$ . At this level, $0^0$ is treated as undefined.

Negative indices

A negative index means take the reciprocal of the corresponding positive power. The reciprocal of a non-zero number $k$ is $1k\frac{1}{k}$ .

a−n=1an,a≠0.a^{-n}=\frac{1}{a^n},\qquad a\neq 0.

More generally,

(ab)−n=(ba)n,\left(\frac{a}{b}\right)^{-n} =\left(\frac{b}{a}\right)^n,

provided all denominators are non-zero.

Key Idea

Meaning of a negative index

A negative index does not make the value negative. It tells you to invert the base:

a−n=1an.a^{-n}=\frac{1}{a^n}.

Example

Removing negative indices

Write $6x−2y−3\frac{6x^{-2}}{y^{-3}}$ using positive indices only, where $x≠0x\neq 0$ and $y≠0y\neq 0$ .

Rewrite $x^{-2}$ as its reciprocal:
$x−2=1x2.x^{-2}=\frac{1}{x^2}.$
Dividing by $y^{-3}$ is equivalent to multiplying by its reciprocal, $y^3$ .
Therefore,
$6x−2y−3=6⋅1x2⋅y3=6y3x2.\frac{6x^{-2}}{y^{-3}} =6\cdot\frac{1}{x^2}\cdot y^3 =\frac{6y^3}{x^2}.$

Tip

Moving factors across a fraction bar

A factor with a negative index becomes a positive-index factor when moved across the fraction bar. For example, $x−3y−2=y2x3\frac{x^{-3}}{y^{-2}}=\frac{y^2}{x^3}$ .

Fractional indices

A fractional index represents a root. For a positive integer $n$ ,

a1/n=an.a^{1/n}=\sqrt[n]{a}.

The denominator of the index tells you which root to take:

a1/2=a,a1/3=a3.a^{1/2}=\sqrt{a}, \qquad a^{1/3}=\sqrt[3]{a}.

For a general rational exponent,

am/n=(an)m=amn.a^{m/n}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}.

The numerator gives the power and the denominator gives the root.

Example

Evaluating a fractional power

Evaluate $81^{3/4}$ .

Interpret the denominator 4 as a fourth root:
$813/4=(814)3.81^{3/4}=\left(\sqrt[4]{81}\right)^3.$
Since $81=3^4$ , its positive fourth root is 3.
Raise this result to the power 3:
$81^{3/4}=3^3=27.$

Example

Combining negative and fractional indices

Evaluate $32^{-2/5}$ .

Use the negative index to form a reciprocal:
$32−2/5=1322/5.32^{-2/5}=\frac{1}{32^{2/5}}.$
Use the denominator 5 to take the fifth root: $325=2\sqrt[5]{32}=2$ .
Use the numerator 2 as the power:
$32^{2/5}=2^2=4.$
Therefore,
$32−2/5=14.32^{-2/5}=\frac{1}{4}.$

Restrictions for rational exponents

When working only with real numbers, even roots of negative numbers are not defined. Therefore, an expression such as $a^{1/2}$ requires $a≥0a\geq 0$ .

Odd roots of negative numbers are real. For example,

8)^{1/3}=-2.

If a rational exponent is negative, the base must also be non-zero because the result involves a reciprocal.

Common Mistake

Check the base and denominator

For arbitrary rational indices, the index laws are most straightforward when bases are positive. With negative bases, first write the exponent in its simplest fractional form and check that any required root is real. For example, $16)^{1/2}$ is not real, while $8)^{2/3}=4$ .

Simplifying a full expression

Many questions require several laws in the same calculation. Work with coefficients and each base separately.

Example

Using several index laws

Simplify

(2x3/2y−1)24x−1y3,\frac{\left(2x^{3/2}y^{-1}\right)^2}{4x^{-1}y^3},

using positive indices, where $x > 0$ and $y≠0y\neq 0$ .

Apply the power to each factor in the numerator:
$(2x3/2y−1)2=4x3y−2.\left(2x^{3/2}y^{-1}\right)^2 =4x^3y^{-2}.$
Divide the coefficients and subtract the indices of each common base:
$4x3y−24x−1y3=x3−(−1)y−2−3=x4y−5.\frac{4x^3y^{-2}}{4x^{-1}y^3} =x^{3-(-1)}y^{-2-3} =x^4y^{-5}.$
Replace the negative index with a reciprocal:
$x4y−5=x4y5.x^4y^{-5}=\frac{x^4}{y^5}.$

Solving equations involving indices

Fractional indices can often be removed by raising both sides of an equation to a suitable power. You must then check that your solution satisfies the original equation and any domain restrictions.

Example

Solving a fractional-index equation

Solve $x^{3/2}=64$ for real $x$ .

Because $x^{3/2}$ involves a square root, the real domain requires $x≥0x\geq 0$ .
Raise both sides to the reciprocal power $23\frac{2}{3}$ :
$x=(64)2/3.x=\left(64\right)^{2/3}.$
Evaluate by taking the cube root and then squaring:
$x=(643)2=42=16.x=\left(\sqrt[3]{64}\right)^2=4^2=16.$
Check in the original equation:
$163/2=(16)3=43=64.16^{3/2}=\left(\sqrt{16}\right)^3=4^3=64.$
Therefore, $x = 16$ .

Exam technique

In the exam

Identify whether you should add, subtract or multiply indices: multiplication of like bases means add, division means subtract, and a power of a power means multiply.
Simplify fractional indices carefully: the denominator gives the root and the numerator gives the power.
State restrictions when relevant, especially for denominators, negative indices and even roots, and leave exact answers unless a decimal is requested.
In longer expressions, deal separately with the numerical coefficient and each algebraic base, then rewrite negative indices as positive indices if required.

Self review

Check yourself

Can you simplify $x5/3⋅x−1/2x^{5/3}\cdot x^{-1/2}$ as a single power of $x$ ?
Can you evaluate $27^{-2/3}$ exactly?
What restrictions are needed for the real expression $1x1/2\frac{1}{x^{1/2}}$ ?

What you'll learn

How to apply the laws of indices when multiplying, dividing and raising powers to powers.
What zero, negative and fractional indices mean.
How to work with any rational exponent and state necessary restrictions.
How to simplify expressions and solve equations involving indices.

Starting point: powers and indices

A power is an expression such as $a^n$ . The number or expression $a$ is the base, and $n$ is the index or exponent.

For a positive integer $n$ ,

factors.a^n=\underbrace{a\cdot a\cdot \ldots \cdot a}_{n\text{ factors}}.

For example, $x4=x⋅x⋅x⋅xx^4=x\cdot x\cdot x\cdot x$ .

Definition

Rational exponent

Multiplying powers with the same base

When powers with the same base are multiplied, add their indices:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

This follows from counting the total number of factors of $a$ . For example, $a3⋅a2a^3\cdot a^2$ contains three factors of $a$ followed by another two, giving five factors altogether.

Key Idea

Product law

Keep the base unchanged and add the indices:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

Example

Multiplying powers

Simplify $3x5/2⋅4x7/33x^{5/2}\cdot 4x^{7/3}$ .

Multiply the numerical coefficients: $3⋅4=123\cdot 4=12$ .
Add the indices of the common base $x$ :
$52+73=156+146=296.\frac{5}{2}+\frac{7}{3} =\frac{15}{6}+\frac{14}{6} =\frac{29}{6}.$
Therefore,
$3x5/2⋅4x7/3=12x29/6.3x^{5/2}\cdot 4x^{7/3}=12x^{29/6}.$

Common Mistake

Multiplying the indices

Do not use $am⋅an=amna^m\cdot a^n=a^{mn}$ . The indices are added, not multiplied. Multiplication of indices belongs to the power-of-a-power law.

Dividing powers with the same base

When powers with the same non-zero base are divided, subtract the index in the denominator from the index in the numerator:

aman=am−n,a≠0.\frac{a^m}{a^n}=a^{m-n},\qquad a\neq 0.

The restriction $a≠0a\neq 0$ is needed because division by zero is undefined.

Key Idea

Quotient law

Keep the base unchanged and subtract the indices in the order top minus bottom:

aman=am−n.\frac{a^m}{a^n}=a^{m-n}.

Example

Dividing powers

Simplify

18p7/4q56p1/2q8,\frac{18p^{7/4}q^5}{6p^{1/2}q^8},

where $p≠0p\neq 0$ and $q≠0q\neq 0$ .

Divide the coefficients: $18÷6=318\div 6=3$ .
Subtract the indices of $p$ :
$74−12=74−24=54.\frac{7}{4}-\frac{1}{2} =\frac{7}{4}-\frac{2}{4} =\frac{5}{4}.$
Subtract the indices of $q$ : $5 - 8 = - 3$ .
Hence,
$18p7/4q56p1/2q8=3p5/4q−3=3p5/4q3.\frac{18p^{7/4}q^5}{6p^{1/2}q^8} =3p^{5/4}q^{-3} =\frac{3p^{5/4}}{q^3}.$

Raising a power to another power

When one power is raised to another power, multiply the indices:

(am)n=amn.\left(a^m\right)^n=a^{mn}.

Brackets matter: the outer index applies to the whole expression inside them.

Example

Raising a power to a power

Simplify $(8x6)2/3\left(8x^6\right)^{2/3}$ , assuming $x≥0x\geq 0$ .

Apply the outer index to both factors:
$(8x6)2/3=82/3(x6)2/3.\left(8x^6\right)^{2/3}=8^{2/3}\left(x^6\right)^{2/3}.$
Evaluate the numerical power using the cube root:
$82/3=(83)2=22=4.8^{2/3}=\left(\sqrt[3]{8}\right)^2=2^2=4.$
Multiply the indices on $x$ :
$(x6)2/3=x6⋅2/3=x4.\left(x^6\right)^{2/3}=x^{6\cdot 2/3}=x^4.$
Therefore,
$(8x6)2/3=4x4.\left(8x^6\right)^{2/3}=4x^4.$

Powers of products and quotients

An index can be distributed across multiplication:

ab)^n=a^nb^n.

It can also be distributed across a quotient, provided the denominator is non-zero:

(ab)n=anbn,b≠0.\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\qquad b\neq 0.

Common Mistake

Powers do not distribute over addition

In general, $(a+b)n≠an+bn(a+b)^n\neq a^n+b^n$ . For example, $2+3)^2=25$ , but $2^2+3^2=13$ . The distribution laws apply to products and quotients, not sums or differences.

The zero index

For any non-zero base,

a^0=1.

This follows from the quotient law:

amam=am−m=a0.\frac{a^m}{a^m}=a^{m-m}=a^0.

But the fraction on the left equals 1, so $a^0=1$ .

Common Mistake

The base cannot be zero

The zero-index law requires $a≠0a\neq 0$ . At this level, $0^0$ is treated as undefined.

Negative indices

A negative index means take the reciprocal of the corresponding positive power. The reciprocal of a non-zero number $k$ is $1k\frac{1}{k}$ .

a−n=1an,a≠0.a^{-n}=\frac{1}{a^n},\qquad a\neq 0.

More generally,

(ab)−n=(ba)n,\left(\frac{a}{b}\right)^{-n} =\left(\frac{b}{a}\right)^n,

provided all denominators are non-zero.

Key Idea

Meaning of a negative index

A negative index does not make the value negative. It tells you to invert the base:

a−n=1an.a^{-n}=\frac{1}{a^n}.

Example

Removing negative indices

Write $6x−2y−3\frac{6x^{-2}}{y^{-3}}$ using positive indices only, where $x≠0x\neq 0$ and $y≠0y\neq 0$ .

Rewrite $x^{-2}$ as its reciprocal:
$x−2=1x2.x^{-2}=\frac{1}{x^2}.$
Dividing by $y^{-3}$ is equivalent to multiplying by its reciprocal, $y^3$ .
Therefore,
$6x−2y−3=6⋅1x2⋅y3=6y3x2.\frac{6x^{-2}}{y^{-3}} =6\cdot\frac{1}{x^2}\cdot y^3 =\frac{6y^3}{x^2}.$

Tip

Moving factors across a fraction bar

A factor with a negative index becomes a positive-index factor when moved across the fraction bar. For example, $x−3y−2=y2x3\frac{x^{-3}}{y^{-2}}=\frac{y^2}{x^3}$ .

Fractional indices

A fractional index represents a root. For a positive integer $n$ ,

a1/n=an.a^{1/n}=\sqrt[n]{a}.

The denominator of the index tells you which root to take:

a1/2=a,a1/3=a3.a^{1/2}=\sqrt{a}, \qquad a^{1/3}=\sqrt[3]{a}.

For a general rational exponent,

am/n=(an)m=amn.a^{m/n}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}.

The numerator gives the power and the denominator gives the root.

Example

Evaluating a fractional power

Evaluate $81^{3/4}$ .

Interpret the denominator 4 as a fourth root:
$813/4=(814)3.81^{3/4}=\left(\sqrt[4]{81}\right)^3.$
Since $81=3^4$ , its positive fourth root is 3.
Raise this result to the power 3:
$81^{3/4}=3^3=27.$

Example

Combining negative and fractional indices

Evaluate $32^{-2/5}$ .

Use the negative index to form a reciprocal:
$32−2/5=1322/5.32^{-2/5}=\frac{1}{32^{2/5}}.$
Use the denominator 5 to take the fifth root: $325=2\sqrt[5]{32}=2$ .
Use the numerator 2 as the power:
$32^{2/5}=2^2=4.$
Therefore,
$32−2/5=14.32^{-2/5}=\frac{1}{4}.$

Restrictions for rational exponents

When working only with real numbers, even roots of negative numbers are not defined. Therefore, an expression such as $a^{1/2}$ requires $a≥0a\geq 0$ .

Odd roots of negative numbers are real. For example,

8)^{1/3}=-2.

If a rational exponent is negative, the base must also be non-zero because the result involves a reciprocal.

Common Mistake

Check the base and denominator

Simplifying a full expression

Many questions require several laws in the same calculation. Work with coefficients and each base separately.

Example

Using several index laws

Simplify

(2x3/2y−1)24x−1y3,\frac{\left(2x^{3/2}y^{-1}\right)^2}{4x^{-1}y^3},

using positive indices, where $x > 0$ and $y≠0y\neq 0$ .

Apply the power to each factor in the numerator:
$(2x3/2y−1)2=4x3y−2.\left(2x^{3/2}y^{-1}\right)^2 =4x^3y^{-2}.$
Divide the coefficients and subtract the indices of each common base:
$4x3y−24x−1y3=x3−(−1)y−2−3=x4y−5.\frac{4x^3y^{-2}}{4x^{-1}y^3} =x^{3-(-1)}y^{-2-3} =x^4y^{-5}.$
Replace the negative index with a reciprocal:
$x4y−5=x4y5.x^4y^{-5}=\frac{x^4}{y^5}.$

Solving equations involving indices

Fractional indices can often be removed by raising both sides of an equation to a suitable power. You must then check that your solution satisfies the original equation and any domain restrictions.

Example

Solving a fractional-index equation

Solve $x^{3/2}=64$ for real $x$ .

Because $x^{3/2}$ involves a square root, the real domain requires $x≥0x\geq 0$ .
Raise both sides to the reciprocal power $23\frac{2}{3}$ :
$x=(64)2/3.x=\left(64\right)^{2/3}.$
Evaluate by taking the cube root and then squaring:
$x=(643)2=42=16.x=\left(\sqrt[3]{64}\right)^2=4^2=16.$
Check in the original equation:
$163/2=(16)3=43=64.16^{3/2}=\left(\sqrt{16}\right)^3=4^3=64.$
Therefore, $x = 16$ .

Exam technique

In the exam

Identify whether you should add, subtract or multiply indices: multiplication of like bases means add, division means subtract, and a power of a power means multiply.
Simplify fractional indices carefully: the denominator gives the root and the numerator gives the power.
State restrictions when relevant, especially for denominators, negative indices and even roots, and leave exact answers unless a decimal is requested.
In longer expressions, deal separately with the numerical coefficient and each algebraic base, then rewrite negative indices as positive indices if required.

Self review

Check yourself

Can you simplify $x5/3⋅x−1/2x^{5/3}\cdot x^{-1/2}$ as a single power of $x$ ?
Can you evaluate $27^{-2/3}$ exactly?
What restrictions are needed for the real expression $1x1/2\frac{1}{x^{1/2}}$ ?

What you'll learn

Starting point: powers and indices

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to another power

Powers of products and quotients

The zero index

Negative indices

Fractional indices

Restrictions for rational exponents

Simplifying a full expression

Solving equations involving indices

Review 1.5.1 Laws of indices by teaching Genie

1.5.1 Laws of indices

What you'll learn

Starting point: powers and indices

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to another power

Powers of products and quotients

The zero index

Negative indices

Fractional indices

Restrictions for rational exponents

Simplifying a full expression

Solving equations involving indices

Review 1.5.1 Laws of indices by teaching Genie

1.5.1 Laws of indices

What you'll learn

Starting point: powers and indices

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to another power

Powers of products and quotients

The zero index

Negative indices

Fractional indices

Restrictions for rational exponents

Simplifying a full expression

Solving equations involving indices

Review 1.5.1 Laws of indices by teaching Genie

1.5.1 Laws of indices Revision Guide

1.5.1 Laws of indices

What you'll learn

Starting point: powers and indices

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to another power

Powers of products and quotients

The zero index

Negative indices

Fractional indices

Restrictions for rational exponents

Simplifying a full expression

Solving equations involving indices

Review 1.5.1 Laws of indices by teaching Genie

1.5.1 Laws of indices Revision Guide