POP

C1.3 Powers & Roots

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01 — Explanation

Which squares and cubes do I actually need to know by heart?

Cambridge expects instant recall — without a calculator — of:

Squares from $1$ to $15$ :

1,4,9,16,25,36,49,64,81,100,121,144,169,196,225

Cubes of $1, 2, 3, 4, 5$ and $10$ :

1, 8, 27, 64, 125, 1000

A root simply undoes a power: $\sqrt{169} = 13$ because $13^2 = 169$ , and $\sqrt[3]{27} = 3$ because $3^3 = 27$ .

02 — Explanation

How do roots behave with fractions and sums?

Roots split apart over a fraction (division), but not over a sum:

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \qquad \text{e.g.} \quad \sqrt{\frac{100}{4}} = \frac{\sqrt{100}}{\sqrt{4}} = \frac{10}{2} = 5

\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}

For example, $\sqrt{9+16} = \sqrt{25} = 5$ , which is not the same as $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ . Always work out what's inside the root first.

03 — Explanation

What happens when I square or cube a negative number?

(-6)^2 = (-6) \times (-6) = 36 \qquad \text{(two negatives multiply to a positive)}

(-2)^3 = (-2) \times (-2) \times (-2) = -8 \qquad \text{(three negatives multiply to a negative)}

The rule generalises: an even power of a negative number is positive, and an odd power stays negative.

Calculator Tip

Typing -5^2 on many calculators gives $-25$ , because it squares $5$ first and then applies the minus sign. Always type (-5)^2 with brackets if you want the negative included inside the square.

04 — Explanation

Why does \sqrt{49}=7, but the equation x^2=49 has two solutions?

The square root symbol $\sqrt{\phantom{x}}$ is defined to give only the positive ("principal") root, so $\sqrt{49}=7$ exactly — never $\pm 7$ . But when you solve an equation like $x^2=49$ , you are asking "which numbers, squared, give $49$ ?", and both $7$ and $-7$ satisfy that, since $(-7)^2=49$ too. So the equation has two solutions, $x=7$ or $x=-7$ , even though the symbol $\sqrt{49}$ on its own only ever means $+7$ .

Common Slip

writing $\sqrt{49}=\pm 7$ . The root symbol itself is always positive — the $\pm$ only appears when you are solving an equation, not when evaluating a root directly.

05 — Explanation

How do I use my calculator for powers and roots that aren't in the memorised list?

For anything beyond the squares $1$ – $15$ and cubes of $1,2,3,4,5,10$ , use the $x^{\Box}$ (or $\wedge$ ) button for powers, and the $\sqrt[\Box]{\phantom{x}}$ button for roots of any order — most calculators need you to type the root number first, then the value.

Example: find $6^4$ and $\sqrt[5]{7776}$ .

$6^4$ : press 6, ^, 4, = to get $1296$ .
$\sqrt[5]{7776}$ : press 5, then the general root button, then 7776, = to get $6$ (since $6^5=7776$ ).

Even with a calculator, it's worth sanity-checking the size of the answer using nearby memorised values — e.g. knowing $6^2=36$ and $6^3=216$ tells you $6^4$ should be noticeably bigger than $216$ , so $1296$ is a believable size.

06 — Worked example

Worked exam-style question

Work out the value of $\sqrt{169} - \sqrt[3]{27} + 2^4$ , then find $(-5)^2 - (-2)^3$ .

First expression:

$\sqrt{169} = 13$ (since $13^2 = 169$ ).
$\sqrt[3]{27} = 3$ (since $3^3 = 27$ ).
$2^4 = 16$ .
Combine: $13 - 3 + 16 = 26$ .

Second expression:

$(-5)^2 = 25$ (negative squared is positive).
$(-2)^3 = -8$ (negative cubed stays negative).
Subtracting a negative flips the sign: $25 - (-8) = 25 + 8 = 33$ .

07 — Common mistakes

Common mistakes to avoid

Splitting a root over addition, e.g. treating $\sqrt{9+16}$ as $\sqrt{9}+\sqrt{16}$ . Always add first, then take the root.
Forgetting brackets around a negative base when entering it into a calculator, which silently changes the sign of the answer.
Assuming every power of a negative number is positive. Only even powers flip the sign to positive; odd powers (like cubing) keep it negative.