Why is minus 1.6 squared negative but minus 5 in brackets squared positive?

The power binds more tightly than the sign, so minus 1.6 squared means the negative of 1.6 squared, which is minus 2.56. Only when the negative is inside brackets, as in minus 5 in brackets squared, does the squaring make the result positive.

Sec 1 Maths: Integers & Rational Numbers — practice questions & worked solutions

Q: Are these from real exam papers?

Yes. All these questions come from Singapore secondary school examination papers from 2016 to 2025, each labelled with its source school, year and paper.

Integers and rational-number questions from real Singapore Secondary 1 examination papers (2016–2025), each with a full worked solution that shows every step — the way marks are actually awarded.

About this topic & key methods

Integers and rational numbers are the foundation of Secondary 1 Mathematics. The questions below test the everyday number skills that everything else builds on: operations with negative numbers, the correct order of operations (brackets, indices, roots, then $\times \div$ , then $+\,-$ ), arithmetic with fractions and mixed numbers, and telling apart rational and irrational numbers, natural numbers, primes and perfect squares.

A common trap at this level is sign errors when subtracting a negative or squaring a negative, and mis-ordering recurring decimals on a number line. Each worked solution below keeps one logical step per line so you can see exactly where each value comes from. For a structured programme that drills these number skills from the ground up, see our Secondary 1 Maths tuition.

Key methods

Order of operations: brackets first, then indices and roots, then $\times$ and $\div$ left to right, then $+$ and $–$ .
Subtracting a negative adds: $a-(-b)=a+b$ ; multiplying/dividing two numbers of the same sign gives a positive, of opposite signs gives a negative.
Squaring a negative is positive, e.g. $(-5)^2=25$ ; but $-1.6^2=-(1.6^2)$ because the power binds before the sign.
Fractions: convert mixed numbers to improper fractions, use a common denominator to add/subtract, and “multiply by the reciprocal” to divide.
Rational numbers can be written as a fraction $\tfrac{p}{q}$ — this includes terminating and recurring decimals; irrational numbers (e.g. $\sqrt{5}$ , $\pi$ ) cannot.
Natural numbers are the positive counting numbers; primes have exactly two factors; perfect squares are squares of whole numbers.
Ordering on a number line: convert everything to decimals; the more negative a number, the smaller it is.

Questions & worked solutions

Q1 — Rational & natural numbers from a list

Raffles Girls’ · 2016 · EYA Q1a · ●○○ Foundational · 2 marks

From the given list, write down all the numbers that are

$\sqrt{8},\quad 0,\quad 2\pi,\quad -6,\quad \sqrt{49}$

(i) rational numbers, (ii) natural numbers.

Show worked solution▾

(i) Rational numbers: $0$ , $-6$ , $\sqrt{49}$ .

Since $\sqrt{49}=7$ , and $\sqrt{8}$ and $2\pi$ are non-terminating, non-recurring decimals.

(ii) Natural numbers: $\sqrt{49}$ .

Since $\sqrt{49}=7$ is a positive whole number.

Q2 — Cube of a fraction, leave answer as a fraction

Raffles Girls’ · 2019 · EYA Q2 · ●●○ Standard · 3 marks

Without the use of a calculator, evaluate $\dfrac{5}{4}\left[\left(\dfrac{2}{3}-2\right)^{3}+0.8\right]$ , leaving your answer as a fraction.

Show worked solution▾

\begin{aligned} \frac{2}{3}-2 &= -\frac{4}{3} \end{aligned}

\begin{aligned} \left(-\frac{4}{3}\right)^{3} &= -\frac{64}{27} \end{aligned}

\begin{aligned} -\frac{64}{27}+0.8 &= -\frac{64}{27}+\frac{4}{5} \end{aligned}

\begin{aligned} &= -\frac{320}{135}+\frac{108}{135} \end{aligned}

\begin{aligned} &= -\frac{212}{135} \end{aligned}

\begin{aligned} \frac{5}{4}\times\left(-\frac{212}{135}\right) &= -\frac{1060}{540} \end{aligned}

\begin{aligned} &= -\frac{53}{27} \end{aligned}

Q3 — Mixed numbers, square & cube root

Temasek JC (IP1) · 2020 · IP1-EOY Q3 · ●●○ Standard · 5 marks

Without the use of a calculator, evaluate

$3 – \left[ 2\frac{4}{5} \times \left( -1\frac{3}{7} \right)^{2} – \left( \frac{\sqrt[3]{64}}{3} \right) \div \left( -1\frac{2}{5} \right) \right],$

showing your working clearly.

Show worked solution▾

\begin{aligned} \left( -1\tfrac{3}{7} \right)^{2} &= \left( -\tfrac{10}{7} \right)^{2} = \frac{100}{49} \end{aligned}

\begin{aligned} 2\tfrac{4}{5} \times \frac{100}{49} &= \frac{14}{5} \times \frac{100}{49} = \frac{1400}{245} = \frac{40}{7} \end{aligned}

\begin{aligned} \frac{\sqrt[3]{64}}{3} &= \frac{4}{3} \end{aligned}

\begin{aligned} \frac{4}{3} \div \left( -1\tfrac{2}{5} \right) &= \frac{4}{3} \div \left( -\tfrac{7}{5} \right) = \frac{4}{3} \times \left( -\tfrac{5}{7} \right) = -\frac{20}{21} \end{aligned}

\begin{aligned} \frac{40}{7} – \left( -\frac{20}{21} \right) &= \frac{120}{21} + \frac{20}{21} = \frac{140}{21} = \frac{20}{3} \end{aligned}

\begin{aligned} 3 – \frac{20}{3} &= \frac{9}{3} – \frac{20}{3} = -\frac{11}{3} = -3\frac{2}{3} \end{aligned}

Q4 — Negative index & cube root of a negative

Temasek JC (IP1) · 2021 · IP1-EOY Q1a · ●●○ Standard · 4 marks

Without the use of a calculator, evaluate $\dfrac{18}{-2^4}\times\left(1\dfrac{1}{3}-\dfrac{\sqrt[3]{-64}}{9}\right)$ .

Show worked solution▾

\begin{aligned} \frac{18}{-2^4}\times\left(1\frac{1}{3}-\frac{\sqrt[3]{-64}}{9}\right)&=\frac{18}{-16}\times\left(\frac{4}{3}-\frac{-4}{9}\right) \end{aligned}

\begin{aligned} &=-\frac{9}{8}\times\left(\frac{12}{9}+\frac{4}{9}\right) \end{aligned}

\begin{aligned} &=-\frac{9}{8}\times\frac{16}{9} \end{aligned}

\begin{aligned} &=-2 \end{aligned}

Q5 — Irrational numbers & mixed evaluation

Temasek JC (IP1) · 2022 · IP1-EOY Q1 · ●●○ Standard · 6 marks

(a) Identify all the irrational numbers in the list of numbers below.

$\sqrt{24}, \quad -1.6^2, \quad \frac{22}{7}, \quad \frac{\pi}{3}, \quad \sqrt[3]{343}, \quad 0$

(b) Without the use of a calculator, evaluate

$5 – 2\left[ \frac{\sqrt[3]{125}}{2} + 3 \div 6 \right] + \sqrt{-4^2 + 65}.$

Show worked solution▾

(a) $\sqrt{24}$ is irrational (24 is not a perfect square) and $\dfrac{\pi}{3}$ is irrational ( $\pi$ is irrational).

$\begin{aligned} \text{Irrational numbers: } \sqrt{24}, \ \frac{\pi}{3} \end{aligned}$

( $-1.6^2 = -2.56$ , $\frac{22}{7}$ , $\sqrt[3]{343}=7$ and $0$ are all rational.)

(b)

$\begin{aligned} &5 – 2\left[ \frac{\sqrt[3]{125}}{2} + 3 \div 6 \right] + \sqrt{-4^2 + 65} \\ &= 5 – 2\left[ \frac{5}{2} + \frac{1}{2} \right] + \sqrt{-16 + 65} \\ &= 5 – 2(3) + \sqrt{49} \\ &= 5 – 6 + 7 \\ &= 6 \end{aligned}$

Q6 — Order recurring decimals on a number line

Xinmin · 2017 · EOY-1E Q2 · ●○○ Foundational · 1 mark

Arrange $-3.1\dot{0}\dot{3}$ , $-3\dfrac{103}{1000}$ , $-3$ , $-3.\dot{1}$ in ascending order.

Show worked solution▾

Convert to decimals:

$\begin{aligned} -3.1\dot{0}\dot{3} &= -3.103103\ldots \\ -3\tfrac{103}{1000} &= -3.103 \\ -3 &= -3.000 \\ -3.\dot{1} &= -3.111\ldots \end{aligned}$

More negative is smaller, so ascending order:

$-3.\dot{1},\ -3.1\dot{0}\dot{3},\ -3\tfrac{103}{1000},\ -3$

Q7 — Prime & rational numbers from a list

Xinmin · 2018 · EOY-1E Q3 · ●○○ Foundational · 2 marks

Write down, from the following list,

$\sqrt{7},\ 1.\dot{4},\ \sqrt{4},\ 1,\ \sqrt[3]{-27}$

(a) the prime number(s), (b) the rational number(s).

Show worked solution▾

(a) $\sqrt{4}=2$ , which is prime:

$\sqrt{4}$

(b) $1.\dot{4}$ is a recurring decimal, $\sqrt{4}=2$ , $\sqrt[3]{-27}=-3$ are integers, and $1$ is an integer:

$1.\dot{4},\ \sqrt{4},\ 1,\ \sqrt[3]{-27}$

Losing marks to sign slips and the wrong order of operations? Getting that automatic is exactly what we drill in our
Secondary 1 Maths programme — every method behind these questions, taught step by step.

Frequently asked questions

What is the difference between a rational and an irrational number?▾

A rational number can be written as a fraction

\tfrac{p}{q}

of two integers — this includes all terminating and recurring decimals (e.g.

0.5

1.\dot{4}

\sqrt{49}=7

). An irrational number cannot, so its decimal goes on forever without repeating (e.g.

\sqrt{5}

\pi

\tfrac{\pi}{3}

). See Q1, Q3, Q7 and Q9.

Why is

-1.6^2

negative but

(-5)^2

positive?▾

The power binds more tightly than the sign, so

-1.6^2

means

-(1.6^2)=-2.56

. Only when the negative is inside brackets, as in

(-5)^2

, does the squaring make it positive. This catches many students in Q7.

How do I order negative recurring decimals on a number line?▾

Convert each number to a decimal with enough places to compare, then remember that the further left (more negative) a number is, the smaller it is. Q8 works through exactly this.

Are these from real exam papers?▾

Yes — all these questions come from Singapore secondary school papers (2016–2025), each labelled with its source school, year and paper.

Sec 1 Maths: Integers & Rational Numbers — practice questions & worked solutions

Key methods

Questions & worked solutions

Q1 — Rational & natural numbers from a list

Q2 — Cube of a fraction, leave answer as a fraction

Q3 — Mixed numbers, square & cube root

Q4 — Negative index & cube root of a negative

Q5 — Irrational numbers & mixed evaluation

Q6 — Order recurring decimals on a number line

Q7 — Prime & rational numbers from a list

Frequently asked questions

Related Sec 1 topics