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Using Index Laws

1. Multiplying terms

2. Dividing terms

3. Expressions of the form (xm)n

4. The zero index

5. Negative indices

6. Negative indices

7. Fractional indices

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Multiplying terms

Simplify:

x + x + x + x + x = 5x

Simplify:

x × x × x × x × x = x5

x to the power of 5

x5 as been written using index notation.

xn

The number x is called the base.

The number n is called the index or power.

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We can use index notation to simplify expressions.

For example,

3p × 2p = 3 × p × 2 × p = 6p2

q2 × q3 = q × q × q × q × q = q5

3r × r2 = 3 × r × r × r = 3r3

3t × 3t = (3t)2 or 9t2

Multiplying terms involving indices

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Multiplying terms with the same base

For example,

a4 × a2 = (a × a × a × a) × (a × a)

= a × a × a × a × a × a

= a6

When we multiply two terms with the same base the indices are added.When we multiply two terms with the same base the indices are added.

= a (4 + 2)

In general,

xm × xn = x(m + n)xm × xn = x(m + n)

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Dividing terms

Remember, in algebra we do not usually use the division sign, ÷.

Instead, we write the number or term we are dividing by underneath like a fraction.

For example,

(a + b) ÷ c is written as a + bc

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Like a fraction, we can often simplify expressions by cancelling.

For example,

n3 ÷ n2 =n3

n2

=n × n × n

n × n

= n

6p2 ÷ 3p =6p2

3p

=6 × p × p

3 × p

2

= 2p

Dividing terms

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Dividing terms with the same base

For example,

a5 ÷ a2 =a × a × a × a × a

a × a= a × a × a = a3

4p6 ÷ 2p4 =4 × p × p × p × p × p × p

2 × p × p × p × p= 2 × p × p = 2p2

= a (5 – 2)

= 2p(6 – 4)

When we divide two terms with the same base the indices are subtracted.When we divide two terms with the same base the indices are subtracted.

In general,

xm ÷ xn = x(m – n)xm ÷ xn = x(m – n)

2

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Sometimes terms can be raised to a power and the result raised to another power.

For example,

(y3)2 = (pq2)4 =

Expressions of the form (xm)n

y3 × y3

= (y × y × y) × (y × y × y)

= y6

pq2 × pq2 × pq2 × pq2

= p4 × q (2 + 2 + 2 + 2)

= p4 × q8

= p4q8

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Expressions of the form (xm)n

For example,

(a5)3 = a5 × a5 × a5

= a(5 + 5 + 5)

= a15

When a term is raised to a power and the result raised to another power, the powers are multiplied.When a term is raised to a power and the result raised to another power, the powers are multiplied.

= a(3 × 5)

In general,

(xm)n = xmn(xm)n = xmn

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Expressions of the form (xm)n

Rewrite the following without brackets.

1) (2a2)3 = 8a6 2) (m3n)4 = m12n4

3) (t–4)2 = t–8 4) (3g5)3 = 27g15

5) (ab–2)–2 = a–2b4 6) (p2q–5)–1 = p–2q5

7) (h½)2 = h 8) (7a4b–3)0 = 1

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The zero index

Look at the following division:

y4 ÷ y4 = 1

But using the rule that xm ÷ xn = x(m – n)

y4 ÷ y4 = y(4 – 4) = y0

That means that

y0 = 1

In general, for all x 0,

x0 = 1x0 = 1

Any number or term divided by itself is equal to 1.

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Negative indices

Look at the following division:

b2 ÷ b4 =b × b

b × b × b × b=

1b × b

=1b2

But using the rule that xm ÷ xn = x(m – n)

b2 ÷ b4 = b(2 – 4) = b–2

That means that

b–2 = 1b2

In general,

x–n = 1xn

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Negative indices

Write the following using fraction notation:

u–1 = 1u

2b–4 = 2b4

x2y–3 = x2

y3

This is the reciprocal of u.

2a(3 – b)–2 = 2a

(3 – b)2

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Negative indices

Write the following using negative indices:

2a

=

x3

y4=

p2

q + 2=

3m(n2 + 2)3

=

2a–1

x3y–4

p2(q + 2)–1

3m(n2 + 2)–3

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Indices can also be fractional.

Fractional indices

x × x =12

12 x + =

12

12 x1 = x

But, x × x = x

x1 = x

So, x = x x = x 12

Similarly, x × x × x =13

13

13 x + + =

13

13

13

But, x × x × x = x3 3 3

So, x = x x = x 13 3

The square root of x.

The cube root of x.

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x = x x = x

In general,

Fractional indices

Also, we can write x as x . mn

1n × m

Using the rule that (xm)n = xmn, we can write

1n n

We can also write x as xm × . mn

1n

x × m = (x )m = (x)m1n

1n n

In general,

x = xmx = xm x = (x)mx = (x)mmn n or

mn n

x = (xm) = xm1nm×

n1n

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Here is a summary of the index laws.

xm × xn = x(m + n)

xm ÷ xn = x(m – n)

Index laws

(xm)n = xmn

x1 = x

x0 = 1 (for x = 0)

x = x 1n

n

x = x 12

x = xm or (x)mnmn n

x–1 = 1x

x–n = 1xn