CDS Mathematics · Surds, Indices and Simplification

Laws of Indices

Powers of the same base multiply by adding exponents and nest by multiplying them; when several powers are equal, call the common value k.

All Surds, Indices and Simplification notes CDS Mathematics strategy

Why this matters

Sixteen PYQs, the largest page in the chapter. They fall into three kinds: chains like x = y^a, y = z^b that close up into one exponent equation; the a^x = b^y = c^z family, solved by naming the common value; and comparisons of large powers, solved by raising to a common power.

Concept 1 of 3: The index laws

Everything follows from counting factors:

a^m a^n

m + n

factors,

(a^m)^n

mn

factors. Rewrite every term with one base, add or multiply the exponents, and compare.

Definition

$a^m a^n = a^{m + n}$ , $\dfrac{a^m}{a^n} = a^{m - n}$ , $(a^m)^n = a^{mn}$ , $a^0 = 1$ , $a^{-n} = \dfrac{1}{a^n}$ , $\sqrt[n]{a} = a^{1/n}$ .
If $a^m = a^n$ with $a > 0$ , $a \ne 1$ , then $m = n$ .
Chains: $x = y^a$ , $y = z^b$ , $z = x^c$ give $x = x^{abc}$ , so $abc = 1$ .
Nested radicals: work from the inside, $\sqrt{x\sqrt x} = x^{3/4}$ .

Index laws

a^m a^n = a^{m + n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \dfrac{1}{a^n}

Worked example

Simplify

\left(\dfrac{x^a}{x^b}\right)^{a + b}\left(\dfrac{x^b}{x^c}\right)^{b + c}\left(\dfrac{x^c}{x^a}\right)^{c + a}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q4Moderate

Example 1 · Surds, Indices and Simplification · Laws of Indices

2b = a + c

and

y^2 = xz

, then what is

x^{b-c} y^{c-a} z^{a-b}

equal to?

The base must not be 1

x^{p} = x^{q}

gives

p = q

only if

x \ne 1

(and

x > 0

). That is why the stems add '

x \ne 1

', and why

x = 1

is often an extra solution of an index equation.

Drill 8 more on the index laws

Concept 2 of 3: When several powers are equal

a^x = b^y = c^z

, call the common value

k

. Then

a = k^{1/x}

b = k^{1/y}

c = k^{1/z}

, and any relation between

a

b

c

becomes a relation between the reciprocals of the exponents.

Definition

Let $a^x = b^y = c^z = k$ (with $k \ne 1$ ):

$abc = 1$ gives $\dfrac1x + \dfrac1y + \dfrac1z = 0$ ;
$b^2 = ac$ gives $\dfrac2y = \dfrac1x + \dfrac1z$ ;
$c = ab$ gives $\dfrac1z = \dfrac1x + \dfrac1y$ , so $z = \dfrac{xy}{x + y}$ .
A product of prime powers equal to another fixes the exponents: $43^x 47^y = 43^2 47^2$ gives $x = y = 2$ .

Common value k

a^x = b^y = c^z = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y},\ c = k^{1/z}

Worked example

2^x = 5^y = 10^z

, find

z

in terms of

x

and

y

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q37Moderate

Example 2 · Surds, Indices and Simplification · Laws of Indices

3^x = 4^y = 12^z

, then z is equal to

Reciprocals of the exponents, not the exponents

10 = 2\times 5

gives

\dfrac1z = \dfrac1x + \dfrac1y

, not

z = x + y

. Products of the bases become SUMS of the reciprocal exponents.

Drill 4 more on when several powers are equal

Concept 3 of 3: Comparing powers and roots

To compare

\sqrt2

\sqrt[3]{3}

\sqrt[6]{6}

, raise all of them to the same power so the roots disappear. To compare huge powers, bring them to the same exponent or compare their logarithms.

Definition

For positive numbers, raising to the same positive power keeps the order.
Roots: raise to the LCM of the root indices.
Powers: bring to a common exponent, e.g. $2^{30} = 8^{10}$ and $3^{20} = 9^{10}$ , so $3^{20} > 2^{30}$ .
A larger exponent usually beats a larger base: compare $n\log a$ .

Common exponent

a^{mn} = (a^m)^n

Worked example

Order

\sqrt[3]{4}

\sqrt{3}

and

\sqrt[6]{10}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — Elementary Mathematics · Q20Moderate

Example 3 · Surds, Indices and Simplification · Laws of Indices

x = \sqrt{2}

y = \sqrt[3]{3}

and

z = \sqrt[6]{6}

, then which one of the following is correct ?

Raise every number to the same power

Comparing

\sqrt2

and

\sqrt[3]3

by squaring one and cubing the other proves nothing. Use one exponent, the LCM of the root indices, for all of them.

Drill 1 more on comparing powers and roots

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The index laws
Index laws
$a^m a^n = a^{m + n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \dfrac{1}{a^n}$
When several powers are equal
Common value k
$a^x = b^y = c^z = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y},\ c = k^{1/z}$
Comparing powers and roots
Common exponent
$a^{mn} = (a^m)^n$

Watch out for (3)

The base must not be 1
→ The index laws
Reciprocals of the exponents, not the exponents
→ When several powers are equal
Raise every number to the same power
→ Comparing powers and roots

Test yourself on Surds, Indices and Simplification

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