1.4

Square Roots and Cube Roots

ASquare Roots

Recall that $3 \times 3 = 3^2$ when expressed in index notation. '3²' is read as '3 squared'. Since $3^2 = 9$, 9 is called the square of 3. We also say that 3 is the positive square root of 9 and it is denoted by $\sqrt{9} = 3$.

Similarly, we write

$2^2 = 2 \times 2 = 4$   and   $\sqrt{4} = \sqrt{2^2} = 2$
$4^2 = 4 \times 4 = 16$   and   $\sqrt{16} = \sqrt{4^2} = 4$
$5^2 = 5 \times 5 = 25$   and   $\sqrt{25} = \sqrt{5^2} = 5$

The numbers 1, 4, 9, 16, 25, ... are perfect squares or square numbers. We can find the square root of a perfect square by using prime factorisation.

A number whose square root is a whole number is called a perfect square.

📖 WORKED EXAMPLE 13

Express 144 as the product of its prime factors in index notation. Hence find $\sqrt{144}$.

Analysis

144 is a perfect square — it can be expressed as the product of two identical numbers, or two identical groups of prime factors.

Solution

Express 144 as the product of its prime factors:
$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2$

Split the prime factors into two equal groups:

$144 = (2 \times 2 \times 3) \times (2 \times 2 \times 3)$
$= (2^2 \times 3) \times (2^2 \times 3)$

✍️ Write the RHS as the square of a number.

× = ²

Here, the boxed expression equals $2^2 \times 3$.

Take the square root:

$\therefore \sqrt{144} = \sqrt{(2^2 \times 3)^2} = 2^2 \times 3 = \mathbf{12}$

NOTE: 'RHS' means 'right-hand side' (等式右边).

THINK²

Look at the index of each prime factor of 144 and the square root of 144. What can you say about the index of each prime factor of any perfect square and that of its square root?

EN The indices of the prime factors of a perfect square are all even. When we take the square root, each index is halved. For example, $144 = 2^4 \times 3^2$ (indices 4, 2 — both even); $\sqrt{144} = 2^2 \times 3 = 12$ (indices halved to 2 and 1).

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文完美平方数的每个质因数的指数都是偶数。开方后，每个指数都减半。例：$144 = 2^4 \times 3^2$（指数 4 和 2 都是偶数）；$\sqrt{144} = 2^2 \times 3 = 12$（指数减半为 2 和 1）。

自评：

✏️ TRY IT YOURSELF 13

Express 484 as the product of its prime factors in index notation. Hence find $\sqrt{484}$.

📖 WORKED EXAMPLE 14

The area of a square is 1521 cm². Find the length of a side of the square by using prime factorisation.

Solution

By prime factorisation, $1521 = 3 \times 3 \times 13 \times 13$
Area of a square = Length × Length
Length $= \sqrt{\text{Area}}$
    $= \sqrt{3 \times 3 \times 13 \times 13}$
    $= \sqrt{(3 \times 13)^2}$
    $= 3 \times 13$
    $= 39$ cm

The length of a side of the square is 39 cm.

1521 cm²

✏️ TRY IT YOURSELF 14

The area of a square is 7225 cm². Find the length of a side using prime factorisation.

📖 WORKED EXAMPLE 15

Given a number $n$ such that $40 \times n$ is a perfect square, find the smallest possible integer value of $n$.

Analysis

In order to make $40 \times n$ a perfect square, one value of $n$ can be 40. That is, $(40 \times \mathbf{40})$ is a perfect square. There are other values of $n$ that can make $40 \times n$ a perfect square.

For example, $40 \times \mathbf{n} = 40 \times \mathbf{\color{#c0327a}{40 \times 2 \times 2}} = (40 \times 2) \times (40 \times 2)$. (Here $n = 40 \times 2 \times 2 = 160$.)

Is 40 the smallest possible value of $n$? How can we find out?

We can express 40 as the product of its prime factors to see if we can find a smaller value of $n$ such that $40 \times n$ is a perfect square. This is because we cannot express $n$ as the product of its prime factors directly.

Solution

$40 \times n = 2 \times 2 \times 2 \times 5 \times n$
$= 2 \times 2 \times 5 \times 2 \times n$
$= (2 \times 2 \times 5) \times (2 \times n)$

Hence to find the smallest positive integer value of $n$ where $40 \times n$ is a perfect square, $(2 \times 2 \times 5)$ and $(2 \times n)$ must be identical.

$40 \times \mathbf{n} = (2 \times 2 \times 5) \times (2 \times \mathbf{\color{#c0327a}{2 \times 5}})$
$\therefore n = 2 \times 5 = \mathbf{10}$

Verify: $40 \times 10 = 400 = 20^2$ ✓

THINK²

(1) What are some other values of $n$ that can make $40 \times n$ a perfect square?
(2) Are you convinced that the smallest value of $n$ is $2 \times 5$ to make $40 \times n$ a perfect square? Explain.

EN (1) Other valid $n$ values: any $n$ that makes the product have all-even prime factor exponents. Examples: $n = 10 \cdot k^2$ for any positive integer $k$, so $n = 10, 40, 90, 160, 250, \ldots$
(2) Yes. $40 = 2^3 \times 5$. For $40 \times n$ to be a perfect square, every prime factor's exponent must be even. The 2's exponent is 3 (odd) → must multiply by at least one more 2. The 5's exponent is 1 (odd) → must multiply by at least one more 5. So $n$ must include at least $2^1 \times 5^1 = 10$. Anything less can't fix both odd exponents.

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中文 (1) 其他可行的 $n$：任何能让乘积所有质因数指数都为偶数的 $n$。例如 $n = 10 \cdot k^2$（$k$ 为正整数），即 $n = 10, 40, 90, 160, 250, \ldots$
(2) 是。$40 = 2^3 \times 5$。要让 $40n$ 是完全平方数，每个质因数的指数都必须是偶数。$2$ 的指数是 3（奇）→ 至少再乘一个 2；$5$ 的指数是 1（奇）→ 至少再乘一个 5。所以 $n$ 至少包含 $2^1 \times 5^1 = 10$。比这更小都补不齐两个奇数指数。

自评：

✏️ TRY IT YOURSELF 15

Given a number $k$ such that $24 \times k$ is a perfect square, find the smallest possible integer value of $k$.

BCube Roots

Recall that $2 \times 2 \times 2 = 2^3$ when expressed in index notation. '2³' is read as '2 cubed'. Since $2^3 = 8$, 8 is called the cube of 2. We also say that 2 is the cube root of 8 and it is denoted by $\sqrt[3]{8} = 2$.

Similarly,

$1^3 = 1 \times 1 \times 1 = 1$   and   $\sqrt[3]{1} = \sqrt[3]{1^3} = 1$
$3^3 = 3 \times 3 \times 3 = 27$   and   $\sqrt[3]{27} = \sqrt[3]{3^3} = 3$
$4^3 = 4 \times 4 \times 4 = 64$   and   $\sqrt[3]{64} = \sqrt[3]{4^3} = 4$

The numbers 1, 8, 27, 64, ... are called perfect cubes or cube numbers.

A number whose cube root is a whole number is called a perfect cube.

📖 WORKED EXAMPLE 16

Find the cube root of 216 using prime factorisation.

Solution

Express 216 as the product of its prime factors:
$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3$

Split into three equal groups:

$216 = (2 \times 3) \times (2 \times 3) \times (2 \times 3)$

✍️ Write the RHS as the cube of a number.

× × = ³

Here, the boxed expression equals $2 \times 3$.

Take the cube root:

$\therefore \sqrt[3]{216} = \sqrt[3]{(2 \times 3)^3} = 2 \times 3 = \mathbf{6}$

THINK²

The prime factorisation of a number is $2^6 \times 3^{12} \times 5^{36}$.
(a) Is the number a perfect cube? Why?
(b) Is the number a perfect square? Why?
(c) For a number that is both a perfect square AND a perfect cube, what can you conclude about the index of each prime factor?

EN (a) Yes — every index (6, 12, 36) is divisible by 3, so the number can be written as $(2^2 \times 3^4 \times 5^{12})^3$.
(b) Yes — every index (6, 12, 36) is even, so the number can be written as $(2^3 \times 3^6 \times 5^{18})^2$.
(c) Each prime factor's index must be divisible by both 2 and 3, hence divisible by 6. The number is a perfect 6th power.

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中文 (a) 是。每个指数（6, 12, 36）都能被 3 整除，所以这个数可以写成 $(2^2 \times 3^4 \times 5^{12})^3$。
(b) 是。每个指数都是偶数，所以这个数可以写成 $(2^3 \times 3^6 \times 5^{18})^2$。
(c) 每个质因数的指数必须既能被 2 整除也能被 3 整除，所以必须能被 6 整除——这个数是一个完美的 6 次方。

自评：

✏️ TRY IT YOURSELF 16

Find the cube root of 1728 using prime factorisation.

📖 WORKED EXAMPLE 17

Given a number $m$ such that $540 \times m$ is a perfect cube, find the smallest possible integer value of $m$.

Analysis

We can express 540 as the product of its prime factors to see if we can find a smaller value of $m$ such that $540 \times m$ can be expressed as three identical groups of prime factors.

Solution

$540 = 2^2 \times 3^3 \times 5$

$540 \times m = 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times m$
$= (2 \times 3 \times 5) \times (2 \times 3 \times 3 \times m)$

Hence to find the smallest positive integer value of $m$ where $540 \times m$ is a perfect cube, the second group must equal $(2 \times 3 \times 5)^2$ when combined with the first group:

$2 \times 3 \times 3 \times m = (2 \times 3 \times 5) \times (2 \times 3 \times 5)$

$\therefore m = 2 \times 5 \times 5 = \mathbf{50}$
(算法: $\dfrac{(2\cdot 3\cdot 5)^2}{2\cdot 3^2} = \dfrac{2^2\cdot 3^2\cdot 5^2}{2\cdot 3^2} = 2 \cdot 5^2 = 50$)

Verify: $540 \times 50 = 27\,000 = 30^3$ ✓

THINK²

(1) Are you convinced that the smallest value of $m$ is $2 \times 5 \times 5$ to make $540 \times m$ a perfect cube? Explain.
(2) What do you notice about the index of the prime factors of $540 \times m$ when $m = 50$?

EN (1) Yes. $540 = 2^2 \times 3^3 \times 5$. For $540 \times m$ to be a perfect cube, every prime factor's exponent must be a multiple of 3. The 2's exponent is 2 (need at least one more 2 → multiply by $2^1$); the 3's exponent is 3 (already ✓); the 5's exponent is 1 (need at least 5² more → multiply by $5^2$). Combining, $m$ must include at least $2 \times 5^2 = 50$. Anything less can't push both odd-exponents up to a multiple of 3.
(2) When $m = 50$: $540 \times 50 = 2^3 \times 3^3 \times 5^3 = 30^3$. Every prime factor's index is exactly 3 — so the product is exactly $(2 \times 3 \times 5)^3$.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文 (1) 是。$540 = 2^2 \times 3^3 \times 5$。要让 $540m$ 是完全立方数，每个质因数的指数必须是 3 的倍数。2 的指数是 2（需要再补 1 个 2，即乘 $2^1$）；3 的指数是 3（已经 ✓）；5 的指数是 1（需要再补 2 个 5，即乘 $5^2$）。所以 $m$ 至少要包含 $2 \times 5^2 = 50$。比这更小都凑不齐"指数升到 3 的倍数"。
(2) 当 $m = 50$ 时：$540 \times 50 = 2^3 \times 3^3 \times 5^3 = 30^3$。每个质因数的指数都恰好是 3——乘积正好是 $(2 \times 3 \times 5)^3$。

自评：

✏️ TRY IT YOURSELF 17

Given a number $k$ such that $600 \times k$ is a perfect cube, find the smallest possible integer value of $k$.

📝 PRACTICE EXERCISE 1.4

BASIC MASTERY

Find the values of these square roots using prime factorisation.

(a) $\sqrt{36}$ =

(b) $\sqrt{121}$ =

(c) $\sqrt{196}$ =

(d) $\sqrt{256}$ =

(e) $\sqrt{441}$ =

(f) $\sqrt{676}$ =
Find the values of these cube roots using prime factorisation.

(a) $\sqrt[3]{343}$ =

(b) $\sqrt[3]{512}$ =

(c) $\sqrt[3]{729}$ =

(d) $\sqrt[3]{1331}$ =

(e) $\sqrt[3]{4096}$ =

(f) $\sqrt[3]{8000}$ =

INTERMEDIATE

Find the positive square roots in index notation.

(a) $\sqrt{5^4 \times 7^2}$ =

(b) $\sqrt{2^6 \times 11^{10}}$ =
Find the cube roots in index notation.

(a) $\sqrt[3]{2^3 \times 19^6}$ =

(b) $\sqrt[3]{3^{12} \times 5^9}$ =
(a) Find the HCF of 63 and 117.

(b) Find the positive square root of the HCF in (a).
Find the cube root of the LCM of 24 and 108.
(a) Find the cube of $2^4 \times 5^2$.

(b) Find the positive square root of the result in (a). Leave answer in index notation.
(a) Find the square of $7^6 \times 19^3$.

(b) Find the cube root of the result in (a). Leave answer in index notation.

ADVANCED

The area of a square tin plate is 7056 cm². Find the length of a side of the plate.
The volume of a glass cube is 2197 cm³. Find the length of a side of the cube.
The area of a square frame is 2601 cm². Find the perimeter of the frame.
(a) Find the prime factorisation of 129 600 in index notation.

(b) The speed of a bullet is $\sqrt{129\,600}$ m/s. Express the speed in index notation.
A piece of wire is cut and bent to form the frame of a cube. If the volume of the cube is 10 648 cm³, find

(a) the length of a side of the cube,

(b) the length of the wire used (12 edges).
It is given that 6 is a factor of 5◇32, where ◇ represents a missing digit.

(a) Find all possible values of ◇.

(b) If 5◇32 is a perfect cube, find

(i) the value represented by ◇: (ii) the cube root of the number:

(c) Study all the possible numbers 5◇32 in (a). State two common properties.

EN The three numbers are 5232, 5532, 5832. Common properties (any two valid):
· All are divisible by 6 (also by 2 and by 3 individually).
· All are 4-digit numbers.
· All end in 2 (last digit is 2; hence all even).
· All start with 5 (first digit is 5).

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中文三个数是 5232, 5532, 5832。共同性质（任写两条）：
· 都能被 6 整除（也都能被 2 和 3 整除）。
· 都是 4 位数。
· 末位数字都是 2（都是偶数）。
· 首位数字都是 5。

自评：
Given a number $n$ such that $2160 \times n$ is a perfect square, find the smallest possible integer value of $n$.
Given a number $m$ such that $36 \times m$ is a perfect cube, find the smallest possible integer value of $m$.
What is the greatest perfect square that is a factor of 1575?
Find the smallest value of $n$ such that $600 \times n$ is both a perfect square AND a perfect cube.

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。本章 4 节都已完成——这些题进入跨章节复习池，第 2 章学习前会重新弹出巩固。