Powers with Letters in the Exponent

Split powers with letters: a^(x+1)=a·a^x, a^(x−2)=a^x/a², a^(2x)=(a^x)²; change base inside the power (9^x = 3^(2x) = y²) and write expressions in terms of y = a^x

In the exam for Edexcel International AS Maths · 3 real past-paper questions below

Split the power, then use y

You can already use the laws of indices on numbers like $5^{3}$ . AS questions put a letter in the power, like $5^{x + 2}$ , and ask you to write it in terms of $y$ , where $y = 5^{x}$ .

The trick: read each law backwards, so the $5^{x}$ splits off on its own. Then swap it for $y$ .

Formula

a^{x + k} = a^{k} \times a^{x} a^{x - k} = \frac{a ^{x}}{a ^{k}} a^{m x} = (a^{x})^{m}

$a$ — the base, a fixed positive number.
$k$ — a plain number added to or taken from the power. It becomes its own number, $a^{k}$ .
$m$ — a number multiplying $x$ . It becomes a power of $y$ .

Memory trick

Plus in the power: times a number. Minus in the power: divide by a number. Times in the power: a power of $y$ .

Worked example

1
Given $y = 5^{x}$ , write $\frac{125}{2 5 ^{1 + x}}$ in terms of $y$ , in simplest form.
2
$2 5^{1 + x} = 25 \times 2 5^{x}$ — Plus in the power: split off the number.
3
$2 5^{x} = (5^{2})^{x} = (5^{x})^{2} = y^{2}$ — Change 25 to base 5 first. Then swap $5^{x}$ for $y$ .
4
$\frac{125}{25 y ^{2}}$ — Put it back together.
5
$= \frac{5}{y ^{2}}$ — Cancel the numbers: $125 \div 25 = 5$ .

Common mistake

Writing $5^{x + 2} = y + 2$ . The 2 is a power, so it becomes the number $5^{2} = 25$ , and it multiplies: $5^{x + 2} = 25 y$ .

Common mistake

Changing the base of the letter part wrongly. $2 5^{x}$ is $y^{2}$ , not $5 y$ or $25 y$ . Always write $25 = 5^{2}$ first.

Rule

Mark schemes give an M mark for using a law correctly, then A1 only for the simplest form: work out the numbers ( $25$ , not $5^{2}$ ), and no fractions inside fractions.

Key idea

Split the power with the laws, change every base to the same number, then put $y$ in. Every question of this type is those three moves.

Fraction powers with letters

Next type: two expressions in $x$ with roots and fractions, and you combine them. The answer must be in the form $k x^{n}$ : one number, one power of $x$ .

Steps

1
1. Turn every root into a power: $x = x^{\frac{1}{2}}$ , $3 x = x^{\frac{1}{3}}$ .
2. Move $x$ off the bottom with a minus power: $\frac{3}{x} = 3 x^{- \frac{1}{2}}$ . The number stays where it is.
3. Deal with the numbers and the powers of $x$ separately.

Worked example

1
$p = 27 x^{6}$ and $q = \frac{3}{x}$ . Write $p^{\frac{1}{3}}$ , $(pq)^{- 1}$ and $\frac{p}{q ^{2}}$ in the form $k x^{n}$ .
2
$q = 3 x^{- \frac{1}{2}}$ — Root to a power, then up with a minus sign. The 3 does not move.
3
$p^{\frac{1}{3}} = 2 7^{\frac{1}{3}} x^{6 \times \frac{1}{3}} = 3 x^{2}$ — The power reaches the number and the $x$ . Cube root of 27 is 3.
4
$pq = 27 \times 3 \times x^{6 - \frac{1}{2}} = 81 x^{\frac{11}{2}}$ — Multiply the numbers, add the powers.
5
$(pq)^{- 1} = \frac{1}{81} x^{- \frac{11}{2}}$ — Power $- 1$ : the number becomes $\frac{1}{81}$ , the power changes sign.
6
$q^{2} = 9 x^{- 1}$ — Square the 3 and double the power.
7
$\frac{p}{q ^{2}} = \frac{27}{9} x^{6 - (- 1)} = 3 x^{7}$ — Divide the numbers, subtract the powers. Bracket the $- 1$ .

Common mistake

Writing $\frac{3}{x} = 3^{- 1} x^{- \frac{1}{2}}$ . Only the part on the bottom gets the minus power. The 3 is on top, so it stays 3.

Common mistake

Leaving $(pq)^{- 1}$ as $81 x^{- \frac{11}{2}}$ . The $- 1$ reaches the 81 too. $k x^{n}$ means $k = \frac{1}{81}$ here.

Key idea

Numbers follow number rules, powers of $x$ follow the index laws. Keep them in two separate lanes and the form $k x^{n}$ drops out.

Split a fraction into powers

Before you can differentiate or integrate a fraction like $\frac{6 x ^{2} + x}{3 3 x}$ , it must be written as separate powers of $x$ : the form $A x^{p} + B x^{q}$ .

Formula

\frac{P + Q}{R} = \frac{P}{R} + \frac{Q}{R}

$P$ — the first term on top.
$Q$ — the second term on top.
$R$ — one single term on the bottom. Each top term is divided by it.

Worked example

1
Write $\frac{6 x ^{2} + x}{3 3 x}$ in the form $A x^{p} + B x^{q}$ .
2
$\frac{6 x ^{2} + x ^{\frac{1}{2}}}{3 x ^{\frac{1}{3}}}$ — Roots to powers first.
3
$\frac{6 x ^{2}}{3 x ^{\frac{1}{3}}} + \frac{x ^{\frac{1}{2}}}{3 x ^{\frac{1}{3}}}$ — Split: each top term over the whole bottom.
4
$2 x^{2 - \frac{1}{3}} = 2 x^{\frac{5}{3}}$ — First part: $6 \div 3 = 2$ , then subtract the powers.
5
$\frac{1}{3} x^{\frac{1}{2} - \frac{1}{3}} = \frac{1}{3} x^{\frac{1}{6}}$ — Second part: $1 \div 3 = \frac{1}{3}$ . $\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$ .
6
$2 x^{\frac{5}{3}} + \frac{1}{3} x^{\frac{1}{6}}$ — So $A = 2$ , $p = \frac{5}{3}$ , $B = \frac{1}{3}$ , $q = \frac{1}{6}$ .

A typical 4-mark question: M1 for changing a root to a power, M1 for using the subtraction law, then A1 for each correct term.

Common mistake

Splitting the bottom instead: $\frac{P}{R + S}$ is not $\frac{P}{R} + \frac{P}{S}$ . You may only split when the top has the plus sign.

Common mistake

Forgetting the number on the bottom for the second term. The 3 divides both parts, so $B = \frac{1}{3}$ , not 1.

Key idea

One term on the bottom? Split over it, then subtract powers. Now each term is ready for the power rule.

Letters in the powers

Some questions give an equation with letters in the powers and ask for $y$ in terms of $x$ .

Write both sides as powers of the same base. Then the powers must be equal, and that gives an ordinary equation.

Worked example

1
Given $\frac{4 ^{x}}{2 ^{y}} = 162$ , find $y$ in terms of $x$ .
2
$4^{x} = 2^{2 x}$ — Change 4 to base 2.
3
$162 = 2^{4} \times 2^{\frac{1}{2}} = 2^{\frac{9}{2}}$ — Change the right side to base 2 as well.
4
$2^{2 x - y} = 2^{\frac{9}{2}}$ — Divide on the left: subtract the powers.
5
$2 x - y = \frac{9}{2}$ — Same base on both sides, so the powers are equal.
6
$y = 2 x - \frac{9}{2}$ — Rearrange.

Common mistake

Multiplying the powers: $2^{4} \times 2^{\frac{1}{2}}$ is $2^{\frac{9}{2}}$ , not $2^{2}$ . Same base multiplied: add the powers.

Key idea

Same base on both sides, then set the powers equal. The letters come down out of the powers and you just rearrange.

Undo a fraction power

Now the letter sits under a power. Get it alone on one side, then raise both sides to the flipped power. The two powers multiply to 1.

Formula

T^{\frac{m}{n}} = c \Rightarrow T = c^{\frac{n}{m}}

$T$ — the letter you want, on its own on one side.
$m$ — top of the power: becomes a root on the other side.
$n$ — bottom of the power: becomes a power on the other side.

Memory trick

To undo a fraction power, flip it: $T^{\frac{3}{2}} = 27$ gives $T = 2 7^{\frac{2}{3}} = 9$ , because $\frac{3}{2} \times \frac{2}{3} = 1$ .

In a model, use the given facts to find the constants first. Then undo the powers in reverse order: plus and minus first, then the power.

Worked example

1
A round patch of moss has radius $r$ cm, $t$ weeks after it is first seen. The model is $r^{3} = a t^{2} + b$ . When $t = 1$ , $r = 2$ . When $t = 3$ , $r = 4$ . Find $a$ and $b$ , then find when the radius reaches 6 cm.
2
$2^{3} = a (1)^{2} + b \Rightarrow a + b = 8$ — Put the first fact into the model.
3
$4^{3} = a (3)^{2} + b \Rightarrow 9 a + b = 64$ — Put in the second fact.
4
$8 a = 56 \Rightarrow a = 7, b = 1$ — Subtract the equations to remove $b$ .
5
$6^{3} = 7 t^{2} + 1 \Rightarrow 7 t^{2} = 215$ — Now $r = 6$ : $6^{3} = 216$ .
6
$t^{2} = \frac{215}{7}$ — Get $t^{2}$ alone.
7
$t = (\frac{215}{7})^{\frac{1}{2}} = 5.54 \dots \approx 5.5$ — Power $\frac{1}{2}$ is a square root. Time is positive. About 5.5 weeks.

Common mistake

Taking the cube root of each term: $r^{3} = a t^{2} + b$ does not give $r = a^{\frac{1}{3}} t^{\frac{2}{3}} + b^{\frac{1}{3}}$ . To find $r$ , work out the whole right side first, then cube root it: $r = (a t^{2} + b)^{\frac{1}{3}}$ .

Key idea

Johannes Kepler found in 1619 that a planet's year $T$ and its distance $a$ from the Sun follow $T^{2} = a^{3}$ (in Earth years and Earth distances). Mars is about 1.52 times as far out, so $T = 1.5 2^{\frac{3}{2}} \approx 1.87$ Earth years.

Key idea

Get the powered letter alone, then flip the power. Round only at the very end.

Which method? Read the question

Look at where the letter sits and what form the answer must take.

You see	You do
Given $y = a^{x}$ , write in terms of $y$	Split the power, match the base, put in $y$
Two expressions, “in the form $k x^{n}$ ”	Roots to powers; numbers and powers separately
A fraction with one term below, “in the form $A x^{p} + B x^{q}$ ”	Split over the bottom, subtract powers
Find $y$ in terms of $x$ , letters in the powers	Same base on both sides, set the powers equal
A letter under a fraction power	Get it alone, then flip the power
A model with two given facts	Two equations, solve, then rearrange

Rule

“You must show all stages of your working” means a calculator answer alone scores nothing. Write each law you use on its own line.

Key idea

Every question here is the index laws read in both directions. Write powers, not roots, and the method is clear.

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