Powers and roots

Edexcel International GCSE Mathematics A· 1.4· 25 分钟阅读

1. Squares, Cubes and Their Roots (Foundation)★☆☆☆☆⏱ 5 min

Square numbers are the product of an integer multiplied by itself, while cube numbers are the product of an integer multiplied by itself twice. The square root of a number is the non-negative value that when squared gives the original number, and the cube root is the value that when cubed gives the original number (can be negative for negative inputs).

📘 定义

Square and Cube Rules

Square number: $n^{2} = n \times n$ for integer $n$ . Square root: $n^{2} = ∣ n ∣$ . Cube number: $n^{3} = n \times n \times n$ for integer $n$ . Cube root: $3 n^{3} = n$ .

📐 例题

Calculate (a) $1 2^{2}$ , (b) $169$ , (c) $5^{3}$ , (d) $3 - 27$

1
(a) Multiply 12 by itself: $12 \times 12 = 144$
2
(b) Find the positive integer that squares to 169: $1 3^{2} = 169$ , so $169 = 13$
3
(c) Multiply 5 by itself three times: $5 \times 5 \times 5 = 125$
4
(d) Find the integer that cubes to -27: $(- 3)^{3} = - 27$ , so $3 - 27 = - 3$

Exam tip:

Memorize the first 15 square numbers and first 10 cube numbers to save calculation time in the exam.

2. Index Laws for Integer Powers (All Tiers)★★☆☆☆⏱ 7 min

Index notation uses a base and a power (index) to represent repeated multiplication efficiently. The index laws let you simplify expressions with powers without expanding them fully, saving calculation time.

📘 定义

Integer Index Laws

For any non-zero base $x$ :

$x^{m} \times x^{n} = x^{m + n}$
$x^{m} \div x^{n} = x^{m - n}$
$(x^{m})^{n} = x^{mn}$
$x^{0} = 1$
$x^{- n} = \frac{1}{x ^{n}}$

📐 例题

Simplify and evaluate (a) $3^{4} \times 3^{2}$ , (b) $7^{5} \div 7^{3}$ , (c) $(2^{3})^{4}$ , (d) $9^{0}$ , (e) $4^{- 2}$

1
(a) Apply multiplication index law: $3^{4 + 2} = 3^{6} = 729$
2
(b) Apply division index law: $7^{5 - 3} = 7^{2} = 49$
3
(c) Apply power of a power index law: $2^{3 \times 4} = 2^{12} = 4096$
4
(d) Apply zero index rule: $9^{0} = 1$
5
(e) Apply negative index rule: $4^{- 2} = \frac{1}{4 ^{2}} = \frac{1}{16}$

Exam tip:

Always confirm the base of the powers is identical before applying multiplication or division index laws.

3. Prime Factorisation, HCF and LCM (All Tiers)★★☆☆☆⏱ 6 min

Any positive integer can be written as a product of prime numbers raised to powers, called its prime factor form. This form is the fastest way to calculate the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers.

📘 定义

HCF and LCM Rules

HCF: Product of the lowest power of each prime factor common to all numbers. LCM: Product of the highest power of every prime factor present in any of the numbers.

📐 例题

Write 72 and 90 as products of prime factors, then calculate their HCF and LCM.

1
Prime factorise 72 using factor tree: $72 = 2^{3} \times 3^{2}$
2
Prime factorise 90 using factor tree: $90 = 2^{1} \times 3^{2} \times 5^{1}$
3
Calculate HCF: Take lowest power of common primes (2, 3): $2^{1} \times 3^{2} = 2 \times 9 = 18$
4
Calculate LCM: Take highest power of all primes (2, 3, 5): $2^{3} \times 3^{2} \times 5^{1} = 8 \times 9 \times 5 = 360$

Exam tip:

Use factor trees to break down large numbers into prime factors quickly, and cross-check your HCF divides both input numbers fully.

4. Fractional and Negative Indices (Higher Only)★★★☆☆Higher 专属⏱ 7 min

For Higher tier, you will need to evaluate expressions with fractional indices, which combine powers and roots. The denominator of the fractional index is the root, and the numerator is the power you raise the result to.

📘 定义

Fractional Index Rule

For positive base $x$ : $x^{\frac{a}{b}} = (b x)^{a} = b x^{a}$ . Combine with the negative index rule for expressions of the form $x^{- \frac{a}{b}} = \frac{1}{x ^{\frac{a}{b}}}$ .

📐 例题

Evaluate (a) $8^{\frac{2}{3}}$ , (b) $62 5^{- \frac{1}{2}}$ , (c) $(\frac{1}{25})^{\frac{3}{2}}$

1
(a) Take cube root first then square: $(38)^{2} = 2^{2} = 4$
2
(b) Apply negative index rule first, then square root: $\frac{1}{625} = \frac{1}{25}$
3
(c) Take square root first then cube: $(\frac{1}{25})^{3} = (\frac{1}{5})^{3} = \frac{1}{125}$

Exam tip:

Always compute the root first for fractional indices to keep numbers small and avoid arithmetic errors.

5. Surds and Rationalising Denominators (Higher Only)★★★☆☆Higher 专属⏱ 7 min

A surd is an irrational root of a positive integer, e.g. $2$ or $7$ . You will need to simplify surds and rationalise denominators (remove surds from the bottom of fractions) to give exact answers, rather than approximate decimal values.

📘 定义

Surd Rules

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
To rationalise a single surd denominator: Multiply numerator and denominator by the surd.
To rationalise $a + b\sqrt{c}$ denominator: Multiply numerator and denominator by the conjugate $a - b\sqrt{c}$.

📐 例题

(a) Simplify $8 + 332$ , (b) Rationalise $\frac{2}{7}$ , (c) Rationalise $\frac{3}{2 + 5}$

1
(a) Simplify each surd first: $8 = 22$ , $332 = 3 \times 42 = 122$ . Add like terms: $22 + 122 = 142$
2
(b) Multiply numerator and denominator by $7$ : $\frac{2 7}{7 \times 7} = \frac{2 7}{7}$
3
(c) Multiply numerator and denominator by conjugate $2 - 5$ : $\frac{3 ( 2 - 5 )}{( 2 + 5 ) ( 2 - 5 )} = \frac{6 - 3 5}{4 - 5} = - 6 + 35$

Exam tip:

Never leave a surd in the denominator unless the question explicitly allows it, or you will lose marks.

6. 常见陷阱

错误做法:

Applying multiplication/division index laws to expressions with different bases, e.g. $2^{3} \times 3^{2} = 6^{5}$

原因:

Index operations only work when the base of all terms is identical

正确做法:

Calculate each term separately before multiplying: $8 \times 9 = 72$

错误做法:

Writing $- 16 = - 4$ , assuming negative numbers have real square roots

原因:

Squares of real numbers are always non-negative, so square roots of negative numbers are not real at this level

正确做法:

Only calculate cube roots of negative numbers; leave square roots of negative numbers as undefined for this syllabus

错误做法:

Calculating the power first for fractional indices leading to very large numbers, e.g. $8^{5/3} = 3276 8^{1/3}$

原因:

Calculating the power first creates unnecessarily large values that are hard to work with

正确做法:

Compute the root first to get a smaller number: $(38)^{5} = 2^{5} = 32$

错误做法:

Using only common prime factors to calculate LCM, e.g. LCM of 72 and 90 = $2^{3} \times 3^{2} = 72$

原因:

LCM must be divisible by both input numbers, so needs all primes from both factorisations

正确做法:

Use the highest power of every prime present in either number: LCM of 72 and 90 = $2^{3} \times 3^{2} \times 5 = 360$

错误做法:

Rationalising $\frac{1}{2 + 3}$ by multiplying only by $3$

原因:

This leaves a cross term with a surd still in the denominator

正确做法:

Multiply numerator and denominator by the conjugate $2 - 3$ to eliminate the surd via difference of squares

7. 速查表

Concept	Rule	Tier
Square/Cube Calculations	$n^{2} = n \times n$ , $n^{2} = ∣ n ∣$ , $n^{3} = n \times n \times n$ , $3 n^{3} = n$	Foundation
Integer Index Laws	$x^{m} \times x^{n} = x^{m + n}$ , $x^{m} \div x^{n} = x^{m - n}$ , $(x^{m})^{n} = x^{mn}$ , $x^{0} = 1$ , $x^{- n} = 1/ x^{n}$	Foundation
HCF via Prime Factors	Product of lowest power of common primes	Foundation
LCM via Prime Factors	Product of highest power of all primes present	Foundation
Fractional Indices	$x^{a / b} = (b x)^{a}$	Higher
Surd Simplification	$ab = a \times b$	Higher
Rationalise Denominator	Multiply by surd (single) or conjugate ( $a + b c$ )	Higher

8. 常见问题

Do I get given the index laws in the exam formula sheet?

No, you must memorize all 5 index laws for both Foundation and Higher tier exams.

When do I need to rationalise a surd denominator?

Unless the question explicitly states otherwise, always rationalise denominators to get full marks for Higher tier surd questions.

Can square roots of negative numbers be calculated for this syllabus?

No, real square roots only exist for non-negative numbers. Negative numbers only have real cube roots at this level.

深入阅读

下一步

Now you have mastered powers and roots for Edexcel IGCSE Maths A, you can move on to algebraic index manipulation (section 2.1 of the specification), which applies the same index laws you have learned to expressions with variables. You should also practice applying these rules to real-world arithmetic problems, and work through tier-specific past paper questions to build speed and accuracy. For Higher tier students, make sure you are confident with surd manipulation before moving on to quadratic formula problems, which often require simplified exact surd answers.

→Quadratic Equations (Higher)