Calculating with Powers and Roots

Powers (or Indices)

A power is a shorthand for repeated multiplication. In the term $a^n$, the base ($a$) is the number being multiplied, and the power ($n$) tells you how many times to multiply it by itself.

For example, $2^5$ means $2 \times 2 \times 2 \times 2 \times 2$, which equals 32.

Special Powers:

• Squared ($x^2$): A number to the power of 2, like $5^2 = 25$. It relates to the area of a square.
• Cubed ($x^3$): A number to the power of 3, like $4^3 = 64$. It relates to the volume of a cube.

Roots

A root is the inverse or “opposite” of a power. It helps you work backwards to find the original base number.

Common Roots:

• Square Root ($\sqrt{x}$): Finds the number that, when multiplied by itself, equals $x$.
  Example: $\sqrt{81} = 9$ because $9 \times 9 = 81$.
• Cube Root ($\sqrt[3]{x}$): Finds the number that, when multiplied by itself three times, equals $x$.
  Example: $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$.

Example 1: Calculate $3^4$

The base is 3 and the power is 4. This means we multiply 3 by itself, 4 times.

$3^4 = 3 \times 3 \times 3 \times 3$

$= 9 \times 3 \times 3$

$= 27 \times 3 = \mathbf{81}$

Example 2: Find $\sqrt{144}$

We need to find a number that, when multiplied by itself, gives 144. We can use our knowledge of perfect squares.

$10^2 = 100$ (too small)

$11^2 = 121$ (close)

$12^2 = 144$ ✓

Answer: 12

Example 3: Find $\sqrt[3]{64}$

We need a number that, when cubed, gives 64. Let’s test some numbers.

$2^3 = 8$ (too small)

$3^3 = 27$ (too small)

$4^3 = 4 \times 4 \times 4 = 64$ ✓

Answer: 4

Example 4: Context Problem

A cube has a volume of $125 \text{ cm}^3$. What is the length of one side?

Volume of a cube $= (\text{side})^3$. To find the side length, we must do the inverse: find the cube root of the volume.

Side $= \sqrt[3]{125} = 5$

Answer: 5 cm

🤔 Common Misunderstandings

• Powers vs. Multiplication: The most common mistake is confusing $5^3$ with $5 \times 3$. Remember, $5^3 = 5 \times 5 \times 5 = 125$, while $5 \times 3 = 15$.
• Forgetting what ‘cubed’ means: Students often remember “squared” for $x^2$ but forget “cubed” means $x^3$.
• Root symbols: Confusing the square root ($\sqrt{x}$) with the cube root ($\sqrt[3]{x}$).

📝 Common Exam Mistakes

• Simple calculation errors: Mistakes when multiplying out a power, e.g., writing $3 \times 3 \times 3 = 26$ instead of 27.
• Not showing working: For multi-mark questions, write out the full multiplication (e.g., $5 \times 5 \times 5$) before the answer.
• Incorrect calculator use: Not knowing where the power ($x^y$ or ^) and root buttons are on the calculator.