Proportionality

Proportion with powers and roots

Advanced Extended 8 min read

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In this note
Formula bank
Writing the equation
Same three steps for every power
What happens when x is multiplied?
Two statements together
Ways they test it
Mistakes to avoid
Summary
Practice
Can I do this?
Quiz

In this note

Syllabus 2.8: proportion with powers and roots Extended

2.8Use proportion to a square, square root, cube or cube root, direct or inverse, including expressions like \((x+3)^2\).
2.8Find the effect on \(y\) of multiplying \(x\) by a number (e.g. \(x\) doubled → \(y\times4\)).
2.8Combine two proportion statements and write \(y\) in terms of another variable.

Before you start: direct and inverse proportion.

Formula bank

Proportional to a power or rootLearn it\(y=kx^{2},\ y=k\sqrt x,\ y=kx^{3},\ y=k\sqrt[3]x\)Use it when "proportional to the square / square root / cube / cube root".

Inversely proportionalLearn it\(y=\dfrac k{x^{2}},\ \dfrac k{\sqrt x},\ \dfrac k{x^{3}}\)Use it when "inversely proportional to the …".

Of a bracketLearn it\(y=k(x+3)^{2}\), \(y=\dfrac k{\sqrt{x+1}}\)the power is on the WHOLE bracket.Use it when "the square of (x + 3)".

Effect of multiplying x by mLearn itdirect: \(y\times m^{n}\); inverse: \(y\div m^{n}\)n is the power in the relationship.Use it when "what happens to y when x is doubled".

Writing the equation

In words	Equation
y is proportional to the square of x	\(y=kx^2\)
y is proportional to the square root of x	\(y=k\sqrt x\)
y is proportional to the cube of x	\(y=kx^3\)
y is proportional to the cube root of x	\(y=k\sqrt[3]x\)
y is inversely proportional to the square of x	\(y=\dfrac k{x^2}\)
y is inversely proportional to the square root of (x + 1)	\(y=\dfrac k{\sqrt{x+1}}\)
y is proportional to the square of (x + 3)	\(y=k(x+3)^2\)

The power applies to the WHOLE bracket the words describe: 'the square of (x + 3)' is \((x+3)^2\), not \(x^2+3\).

Same three steps for every power

General methodHow to answer any proportion question with a power or root

Turn the words into an equation with k (power or root on the whole quantity).
Substitute the given pair and solve for k (work out the power or root first).
Write the equation with the number for k.
Substitute the new value. If you are finding x, undo the power at the end (\(x^{2}=100\Rightarrow x=10\)).

Example \(y\propto x^2\). \(y=50\) when \(x=5\). Find \(y\) when \(x=3\), and \(x\) (positive) when \(y=200\).

\(y=kx^{2}\): \(50=25k\).

\(k=2\), \(y=2x^{2}\)
\(x=3\): \(y=18\)
\(200=2x^{2}\Rightarrow x^{2}=100\)

Answer 18 and 10

Example \(y\propto\dfrac1{\sqrt{x+1}}\). \(y=3\) when \(x=15\). Find \(y\) when \(x=35\).

\(y=\dfrac k{\sqrt{x+1}}\); \(\sqrt{16}=4\).

\(3=\tfrac k4\Rightarrow k=12\)
\(x=35\): \(\sqrt{36}=6\)
\(y=\tfrac{12}6\)

Answer 2

Try it yourself

\(y\propto x^2\), y = 18 at x = 3. y at x = 5?

Show answer

Answer 50

\(y\propto\sqrt x\), y = 12 at x = 9. y at x = 25?

Show answer

Answer 20

\(y\propto\tfrac1{x^2}\), y = 2 at x = 4. y at x = ½?

Show answer

Answer 128

What happens when x is multiplied?

If \(y=kx^2\) and \(x\) is multiplied by 3, then \(y\) becomes \(k(3x)^2=9kx^2\): \(y\) is multiplied by \(3^2=9\).

General methodHow to find the effect of a change in x

Put the multiplier on x through the same power or root (× 4 under a square root → × 2).
Direct: y is multiplied by that. Inverse: y is divided by it.
A percentage change: write it as a multiplier first (+10% → × 1.1).

Relationship	x is multiplied by	y is multiplied by
\(y\propto x^2\)	3	9
\(y\propto\sqrt x\)	4	2
\(y\propto\tfrac1{\sqrt x}\)	4	\(\tfrac12\) (halved)
\(y\propto x^3\)	1.2 (a 20% increase)	1.728 (a 72.8% increase)
\(y\propto\tfrac1{x^2}\)	2	\(\tfrac14\)

Try it yourself

\(y\propto x^3\). x is doubled. y is multiplied by?

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Answer 8

\(y\propto\tfrac1{\sqrt x}\). x × 4. y is multiplied by?

Show answer

Answer ½

Two statements together

General methodHow to combine two proportion statements

Write each as an equation with its own constant.
Substitute one into the other; the constants combine into ONE k.
Find k from the given values.

Example \(y\) is inversely proportional to \(x\), and \(x\) is directly proportional to \(w^2\). When \(w=2\), \(y=3\). Find \(y\) in terms of \(w\).

\(y=\tfrac ax\), \(x=bw^{2}\) → \(y=\tfrac k{w^{2}}\).

\(3=\tfrac k4\Rightarrow k=12\)

Answer \(y=\dfrac{12}{w^2}\)

Ways they test it

Type 1 The standard 3-mark question

"\(y\) is proportional to the square of \((x-1)\). \(y=3\) when \(x=4\). Find \(y\) when \(x=7\)."

Spot it a power or root in words.

Equation (M1).
k (A1).
Answer (A1).

For example \(y=k(x-1)^{2}\)

\(3=9k\Rightarrow k=\tfrac13\)
\(\tfrac13\times36\)

Answer 12

Type 2 Inverse square of a bracket

"\(m\) is inversely proportional to the square of \((t+2)\)…"

Spot it "inversely … square of (…)".

\(m=\dfrac k{(t+2)^{2}}\).

Type 3 Effect of a change

"Explain what happens to p when q is multiplied by 9." (2 marks)

Spot it no numbers for k.

Multiplier through the power, then direct/inverse.

Type 4 Percentage change

"r is increased by 10%. Find the percentage increase in V." (V ∝ r³)

Spot it a % change.

\(1.1^{3}=1.331\) → 33.1%.

Mistakes to avoid

✗ 'square of (x + 3)' → \(x^2+3\)✓ \((x+3)^2\)The whole bracket is squared.
✗ \(y\propto\sqrt x\), x × 4 → y × 4✓ y × 2√4 = 2.
✗ \(200=2x^{2}\) → x = 100✓ x = 10Undo the square.

Summary

Same 3 steps for every power or root: equation, k, use.
The power goes on the whole bracket.
Effects: put the multiplier through the power; divide for inverse.
Chains: one combined k.

If you see …

"square root of (x + 1)"\(\sqrt{x+1}\) in the equation.
"what happens when x is doubled"\(2^{n}\).

Practice

Practice: easy, medium, hard

Easy

\(y\propto x^2\), y = 18 when x = 3. Find k.

Show answer

Answer 2

\(y\propto\sqrt x\), y = 12 when x = 9. Find y when x = 16.

Show working

\(k=4\)
\(4\times4\)

Answer 16

\(y\propto x^2\). x is multiplied by 3. y is multiplied by?

Show answer

Answer 9

Medium

\(y\propto(x+3)^2\). \(y=5\) when \(x=2\). Find \(y\) when \(x=1\).

Guide me

What is it asking? y

What do we know? \(y=k(x+3)^{2}\)

Plan \(5=25k\)

Show working

\(k=\tfrac15\)
\(\tfrac15\times16\)

Answer 3.2

\(y\propto\sqrt[3]{x-2}\). \(y=6\) when \(x=10\). Find \(y\) when \(x=66\).

Guide me

What is it asking? y

What do we know? \(y=k\sqrt[3]{x-2}\)

Plan \(\sqrt[3]8=2\)

Show working

\(k=3\)
\(3\times\sqrt[3]{64}\)

Answer 12

\(y\) is inversely proportional to \(x^2\). \(y=8\) when \(x=3\). Find \(y\) when \(x=6\).

Guide me

What is it asking? y

What do we know? \(y=\tfrac k{x^{2}}\)

Plan k = 8 × 9

Show working

\(k=72\)
\(72\div36\)

Answer 2

Hard

\(m\) is inversely proportional to the square of \((t+2)\). \(m=1.5\) when \(t=2\). Find \(m\) when \(t=6\).

Guide me

What is it asking? m

What do we know? \(m=\dfrac k{(t+2)^{2}}\)

Plan \(1.5=\tfrac k{16}\)

Show working

\(k=24\)
\(\tfrac{24}{64}\)

Answer 0.375

Part	Answer / working	Marks
	equation	M1
	k = 24	A1
	0.375	A1
Total: 3 marks

\(y\) is directly proportional to the square root of \((x+4)\). \(y=6\) when \(x=5\). Find \(y\) when \(x=60\).

Guide me

What is it asking? y

What do we know? \(y=k\sqrt{x+4}\)

Plan \(\sqrt9=3\)

Show working

\(k=2\)
\(2\times\sqrt{64}\)

Answer 16

Part	Answer / working	Marks
	\(y=k\sqrt{x+4}\)	M1
	k = 2	A1
	16	A1
Total: 3 marks

\(V\) is proportional to \(r^3\). \(r\) is increased by 10%. Find the percentage increase in \(V\).

Guide me

What is it asking? a % increase

What do we know? multiplier 1.1

Plan cube it

Show working

\(1.1^{3}=1.331\)

Answer 33.1%

Part	Answer / working	Marks
	\(1.1^3\)	M1
	33.1%	A1
Total: 2 marks

Can I do this?

I can write proportion statements with powers and roots as equations.
I can find k and use it, including with brackets.
I can explain the effect on y of multiplying x.
I can combine two proportion statements.

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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