Powers and Roots

Finding an n-th root is the inverse operation of raising a number to a power. If $a^n = b$, then $\sqrt[n]{b} = a$. The number underneath the root symbol is called the radicand. Even roots (like square roots) of positive numbers always have two real answers (one positive, one negative), but the $\sqrt{}$ symbol specifically refers to the principal root (the positive answer). Negative numbers have no real even roots. Odd roots (like cube roots) behave differently, as every number has exactly one real odd root, which can be positive or negative.

Worked Example: Calculating a power

Evaluate $4^3$.

Step 1: Write out the repeated multiplication using the base (4) and the index (3).

• $4 \times 4 \times 4$

Step 2: Calculate the result sequentially.

• $4 \times 4 = 16$, then $16 \times 4 = 64$.

Answer: $64$

Worked Example: Calculating a root

Evaluate $\sqrt[4]{81}$.

Step 1: Set up the inverse operation. You need a number that multiplies by itself 4 times to make 81.

• $x \times x \times x \times x = 81$

Step 2: Test small integer values.

• $3 \times 3 \times 3 \times 3 = 81$

Answer: $3$

Standard Powers to Recognise

Why do musicians relentlessly practice scales? So they can play them without thinking. In your non-calculator exam, memorising standard integer powers acts as your mathematical muscle memory, allowing you to instantly simplify expressions.

You must be able to state these common powers from memory:

• Square numbers ($x^2$): $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ (Higher Tier must know up to $15^2$: $121, 144, 169, 196, 225$).
• Cube numbers ($x^3$): $1^3=1$, $2^3=8$, $3^3=27$, $4^3=64$, $5^3=125$, and $10^3=1000$.
• Powers of 2: $2^1=2$, $2^2=4$, $2^3=8$, $2^4=16$, $2^5=32$, $2^6=64$.
• Powers of 3: $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$.
• Powers of 4: $4^1=4$, $4^2=16$, $4^3=64$.
• Powers of 5: $5^1=5$, $5^2=25$, $5^3=125$.

Estimating Powers and Roots

What if you need to find the square root of 20 without a calculator? By framing an unknown root between two familiar ones, you can make a highly accurate mathematical guess.

For Edexcel exams, complex estimation usually requires you to round every number to 1 significant figure before calculating. For roots that are surds (irrational roots of positive integers), you must use the bounds method. This involves identifying the nearest perfect squares or cubes above and below the radicand. You can then apply the rule of proximity: if a number is closer to one bound, your decimal estimate should skew toward that bound's root.

Worked Example: Estimating a root

Estimate the value of $\sqrt{20}$.

Step 1: Identify the bounds (the nearest perfect squares below and above 20).

• $16 < 20 < 25$

Step 2: Find the square roots of these perfect squares.

• $\sqrt{16} = 4$ and $\sqrt{25} = 5$

Step 3: Apply the rule of proximity. 20 is roughly halfway between 16 and 25.

Answer: $\approx 4.5$

Worked Example: Complex estimation

Estimate the value of:

$$\frac{4.1^2 \times \sqrt{98.5}}{0.52}$$

Step 1: Round every value to 1 significant figure.

$$\frac{4^2 \times \sqrt{100}}{0.5}$$

Step 2: Simplify the powers and roots.

$$\frac{16 \times 10}{0.5} = \frac{160}{0.5}$$

Step 3: Calculate the final value (remembering that dividing by $0.5$ is exactly the same as multiplying by 2).

• $160 \times 2 = 320$

Answer: $320$