Powers and roots

1. Overview

Powers and roots are inverse mathematical operations. Powers represent repeated multiplication (e.g., $5^3 = 5 \times 5 \times 5$), while roots determine what number, when raised to a power, gives a specified value (e.g., $\sqrt[3]{125} = 5$ because $5^3 = 125$). Mastering powers and roots is crucial for simplifying expressions, solving equations, and tackling problems in geometry, mensuration, and scientific contexts.

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Key Definitions

• Base: The number that is being multiplied by itself (e.g., in $5^3$, 5 is the base).
• Index (Exponent/Power): The number of times the base is multiplied by itself (e.g., in $5^3$, 3 is the index).
• Square: A number multiplied by itself once ($x^2$).
• Square Root ($\sqrt{x}$): The value that, when multiplied by itself, gives the original number.
• Cube: A number multiplied by itself twice ($x^3$).
• Cube Root ($\sqrt[3]{x}$): The value that, when multiplied by itself three times, gives the original number.
• Reciprocal: The result of dividing 1 by the number (e.g., the reciprocal of 5 is $\frac{1}{5}$).

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Core Content

Squares and Square Roots

A square number is the result of $n \times n$, or $n^2$. The square root is the inverse operation, finding the value of $n$ given $n^2$.

• Note: Every positive number has two square roots (one positive and one negative), but the symbol $\sqrt{x}$ usually refers to the "principal" (positive) root. For example, $\sqrt{9} = 3$, although $(-3)^2 = 9$ as well.

Worked example 1 — Square root calculation

Question: Calculate $\sqrt{225}$.

1. Identify the goal: Find a number that, when multiplied by itself, equals 225.
2. Trial/Knowledge: We know $10 \times 10 = 100$. Let's try a larger number. $15 \times 15 = 225$.
3. Result: $\sqrt{225} = \boxed{15}$

Cubes and Cube Roots

A cube number is $n \times n \times n$, or $n^3$. The cube root is the inverse operation.

• Note: You can find the cube root of a negative number (e.g., $\sqrt[3]{-8} = -2$), but you cannot find the square root of a negative number in basic algebra.

Worked example 2 — Cube calculation

Question: Calculate $7^3$.

1. Step 1: $7 \times 7 = 49$
2. Step 2: $49 \times 7 = 343$
3. Result: $7^3 = \boxed{343}$

Other Powers and Roots

You can raise numbers to any power ($n^4, n^5,$ etc.) or find higher roots ($\sqrt[4]{x}, \sqrt[5]{x}$).

• Calculator Tip: Use the $x^y$ or $x^\square$ button for powers and the $\sqrt[x]{\square}$ or shift + $\sqrt{\square}$ for roots.

Worked example 3 — Higher power calculation

Question: Evaluate $3^4$

1. Calculation: $3 \times 3 \times 3 \times 3$
2. Step-by-step: $3 \times 3 = 9 \rightarrow 9 \times 3 = 27 \rightarrow 27 \times 3 = 81$.
3. Result: $3^4 = \boxed{81}$

Worked example 4 — Higher root calculation

Question: Evaluate $\sqrt[4]{16}$

1. Identify the goal: Find a number that, when raised to the power of 4, equals 16.
2. Trial/Knowledge: $1^4 = 1$, $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
3. Result: $\sqrt[4]{16} = \boxed{2}$

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Extended Content (Extended Only)

For the Extended curriculum, you must be able to calculate with fractional and negative indices.

Negative Indices

A negative power indicates a reciprocal. This means $a^{-n}$ is the same as $\frac{1}{a^n}$. A number raised to a negative power is equal to one divided by that number raised to the positive version of the power.

Formula: $\qquad \bf a^{-n} = \frac{1}{a^n}$

Worked example 5 — Negative index

Question: Evaluate $5^{-3}$

1. Method: Use the reciprocal rule.
2. Calculation: $5^{-3} = \frac{1}{5^3}$
3. Simplify: $\frac{1}{5 \times 5 \times 5} = \frac{1}{125}$
4. Result: $5^{-3} = \boxed{0.008}$

Fractional Indices

A fractional index combines a power and a root. The denominator of the fraction indicates the type of root to take, and the numerator indicates the power to raise the result to. For example, $a^{\frac{m}{n}}$ means taking the $n$th root of $a$, and then raising the result to the power of $m$. It's often easier to take the root before raising to the power, to keep the numbers smaller.

Formula: $\qquad \bf a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Worked example 6 — Fractional index

Question: Evaluate $8^{\frac{2}{3}}$

1. Method: Take the cube root first, then square the result.
2. Step 1 (Root): $\sqrt[3]{8} = 2$ (because $2 \times 2 \times 2 = 8$)
3. Step 2 (Power): $2^2 = 4$
4. Result: $8^{\frac{2}{3}} = \boxed{4}$

Worked example 7 — Fractional index with a larger base

Question: Evaluate $64^{\frac{5}{6}}$

1. Method: Take the 6th root first, then raise the result to the power of 5.
2. Step 1 (Root): $\sqrt[6]{64} = 2$ (because $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$)
3. Step 2 (Power): $2^5 = 32$
4. Result: $64^{\frac{5}{6}} = \boxed{32}$

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Key Equations

Concept	Equation	Notes
Multiplication Law	$\bf a^m \times a^n = a^{m+n}$	Add indices when bases are the same.
Division Law	$\bf a^m \div a^n = a^{m-n}$	Subtract indices when bases are the same.
Power of a Power	$\bf (a^m)^n = a^{m \times n}$	Multiply indices.
Zero Index	$\bf a^0 = 1$	Any number (except 0) to the power of 0 is 1.
Fractional Index	$\bf a^{1/n} = \sqrt[n]{a}$	Denominator is the root.

These formulas are NOT provided on the IGCSE formula sheet; they must be memorised.

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Common Mistakes to Avoid

• ❌ Wrong: Thinking $4^3 = 4 \times 3 = 12$. ✓ Right: $4^3 = 4 \times 4 \times 4 = 64$. (A power indicates repeated multiplication of the base by itself).
• ❌ Wrong: Forgetting the order of operations when dealing with negative signs: $-5^2 = 25$. ✓ Right: $-5^2 = -(5 \times 5) = -25$. If you want to square the negative number, use parentheses: $(-5)^2 = (-5) \times (-5) = 25$.
• ❌ Wrong: Assuming $\sqrt{a^2 + b^2} = a + b$. ✓ Right: $\sqrt{a^2 + b^2}$ cannot be simplified in this way. You must first calculate $a^2 + b^2$ and then take the square root of the result. For example, $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
• ❌ Wrong: Confusing negative indices with negative numbers: $2^{-1} = -2$ ✓ Right: A negative index indicates a reciprocal: $2^{-1} = \frac{1}{2} = 0.5$

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Exam Tips

• Command Words:
  • "Evaluate": Find the final numerical value.
  • "Simplify": Leave the answer in index form (e.g., $5^7$).
• Typical Contexts: Powers and roots often appear in questions involving Pythagoras' Theorem ($a^2 + b^2 = c^2$), Compound Interest, and Area/Volume calculations.
• Values to Memorise:
  • Squares: $1^2$ to $15^2$ (e.g., $13^2 = 169$, $14^2 = 196$, $15^2 = 225$).
  • Cubes: $1^3, 2^3, 3^3, 4^3, 5^3,$ and $10^3$.
• Calculator vs Non-Calculator: In non-calculator papers, look for "square numbers in disguise." If you see 49, think $7^2$. If you see 27, think $3^3$.
• Show Your Working: Even if using a calculator, write down the calculation you are performing. If you type it in wrong but wrote down the correct method, you can still get "Method Marks."