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PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i1" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i1"><div class="list-item-paragraph">Find all prime numbers satisfying $-5\leq2x-3\leq7$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i2" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i2"><div class="list-item-paragraph">Simplify $\dfrac{18c^3d^3}{5f^3}\div\dfrac{3c^5d^2}{10f}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i3" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i3"><div class="list-item-paragraph">Simplify $\left(\dfrac{16y^8}{81x^{12}}\right)^{-\frac34}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i4" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i4"><div class="list-item-paragraph">Write $\dfrac{3x}{(2x-1)^2}-\dfrac2{2x-1}$ as a single fraction in its simplest form.<span class="inline-marks">[2]</span></div></li></ol>

Find all prime numbers satisfying $-5\leq2x-3\leq7$.

Simplify $\dfrac{18c^3d^3}{5f^3}\div\dfrac{3c^5d^2}{10f}$.

Simplify $\left(\dfrac{16y^8}{81x^{12}}\right)^{-\frac34}$.

Write $\dfrac{3x}{(2x-1)^2}-\dfrac2{2x-1}$ as a single fraction in its simplest form.

Multiplication and Division of Algebraic Fractions

Solving Simultaneous Linear Inequalities

Prime Numbers and Composite Numbers

Rational Indices

Addition and Subtraction of Algebraic Fractions

2025 YCKSS PRELIMS P2 Q1

Algebraic Expressions, Identities and Formulae

Indices

Linear Equations and Inequalities

Primes, HCF and LCM

<p>Simplify each of the following.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( \frac{{{p}^{3}}}{q} \right)}^{4}}\times {{\left( \frac{3{{q}^{4}}}{{{p}^{3}}} \right)}^{2}}$</div></li><li><div class="list-item-paragraph">$\frac{{{(2{{x}^{5}}{{y}^{4}})}^{2}}}{2{{x}^{8}}{{y}^{2}}}$</div></li><li><div class="list-item-paragraph">${{(-2{{q}^{4}}{{p}^{3}})}^{3}}\div {{(4{{q}^{5}}{{p}^{3}})}^{2}}$ </div></li><li><div class="list-item-paragraph">$2{{a}^{\frac{1}{4}}}{{b}^{\frac{1}{3}}}\div {{(5a)}^{-2}}\times {{b}^{-\frac{2}{3}}}$</div></li></ol>

Laws of Indices

Simplifying Expressions Using Advanced Index Laws

<p>Evaluate without the use of a calculator.		</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{2}^{\frac{3}{2}}}\times \sqrt{8}\div 8$</div></li><li><div class="list-item-paragraph">${{\left( 11\frac{1}{9} \right)}^{-\frac{1}{2}}}$</div></li></ol>

Evaluate without the use of a calculator.		

Laws of Indices: Rational and Mixed Index Applications

<p>Expressing $\sqrt[3]{49}$ as a power of $7$.</p>

Expressing $\sqrt[3]{49}$ as a power of $7$.

Expressing Roots as Powers of a Given Base

<p>Simplify each of the following, leaving your answer in positive indices.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">${{\left( \frac{27}{{{x}^{-3}}} \right)}^{-\mathrm{   }\frac{1}{3}}}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">${{\left( \frac{8{{a}^{-2}}}{\sqrt[3]{b}} \right)}^{-3}}$</div></li></ol></div></div></div>

Simplify each of the following, leaving your answer in positive indices.

Laws of Indices: Positive Form Only

<p>The requested form $x^ay^b$ with fixed integer indices requires a sign condition: for&nbsp;$y&gt;0$, the answer is $x^6y^{-5}$. For arbitrary nonzero real $x,y$, the principal-root answer is $x^6/|y|^5$.</p>
<p>Express ${{\left( \frac{{{x}^{3}}{{y}^{-4}}}{{{x}^{-9}}{{y}^{6}}} \right)}^{\frac{1}{2}}}$ in the form ${{x}^{a}}{{y}^{b}}$, where&nbsp;$a$ and $b$ are integers.  </p>

The requested form $x^ay^b$ with fixed integer indices requires a sign condition: for $y>0$, the answer is $x^6y^{-5}$. For arbitrary nonzero real $x,y$, the principal-root answer is $x^6/|y|^5$.

Express ${{\left( \frac{{{x}^{3}}{{y}^{-4}}}{{{x}^{-9}}{{y}^{6}}} \right)}^{\frac{1}{2}}}$ in the form ${{x}^{a}}{{y}^{b}}$, where $a$ and $b$ are integers.  

Expressing Powers in Simplified Index Form

<p>Express each of the following as a power of $5$. </p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--3" data-layout="3"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">$\sqrt[7]{5}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">$\frac{1}{125}\times \sqrt{5}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="3"><li><div class="list-item-paragraph">$\sqrt[3]{{{125}^{2}}}$</div></li></ol></div></div></div>

Express each of the following as a power of $5$. 

Expressing Roots and Radicals as Powers of 5

<p>Simplify each of the following, expressing your answers in positive index notation.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\sqrt[3]{a}\times 5{{a}^{\frac{5}{3}}}$</div></li><li><div class="list-item-paragraph">${{\left( \sqrt{m} \right)}^{3}}\times {{\left( \sqrt[3]{m} \right)}^{4}}\div {{\left( \sqrt[6]{m} \right)}^{5}}$</div></li><li><div class="list-item-paragraph">$2{{x}^{2}}\times {{\left( \frac{25{{x}^{2}}}{4{{y}^{4}}} \right)}^{\frac{1}{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{{{p}^{2}}}{2{{q}^{4}}}\div \frac{\sqrt{p}}{{{(4{{q}^{3}})}^{2}}}$</div></li><li><div class="list-item-paragraph">${{\left( -3{{m}^{2}}{{n}^{-\frac{2}{3}}} \right)}^{3}}\times \sqrt[3]{8mn}$</div></li></ol>

Simplify each of the following, expressing your answers in positive index notation.

Simplifying Complex Expressions with Rational Indices

<p>Evaluate each of the following without the use of a calculator.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{27}^{\frac{2}{3}}}$</div></li><li><div class="list-item-paragraph">${{16}^{1.5}}$</div></li><li><div class="list-item-paragraph">${{27}^{\frac{1}{3}}}\times {{3}^{-1}}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{3}{4} \right)}^{-2}}\div {{16}^{-\frac{1}{2}}}$</div></li><li><div class="list-item-paragraph">${{8}^{-\frac{2}{3}}}\times {{\left( \frac{2}{5} \right)}^{2}}\div {{5}^{-2}}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{1}{4} \right)}^{0}}-\sqrt{64}+{{\left( \frac{1}{2} \right)}^{-3}}$</div></li></ol>

Evaluate each of the following without the use of a calculator.

Zero and Negative Indices

Evaluating Indices: Fractions, Roots, and Powers

<p>Evaluate each of the following without the use of a calculator.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\sqrt[4]{256}$</div></li><li><div class="list-item-paragraph">$\sqrt[5]{1024}$</div></li><li><div class="list-item-paragraph">$\sqrt[3]{\frac{8}{27}}$</div></li><li><div class="list-item-paragraph">${{49}^{\frac{1}{2}}}$</div></li><li><div class="list-item-paragraph">${{\left( 1\frac{7}{9} \right)}^{-\frac{1}{2}}}$</div></li><li><div class="list-item-paragraph">${{\left( -64 \right)}^{-\frac{1}{3}}}$</div></li></ol>

Roots and Rational Exponents Without a Calculator

<p>Simplify each of the following, leaving your answer in positive indices.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( 27{{x}^{6}} \right)}^{\frac{2}{3}}}$</div></li><li><div class="list-item-paragraph">${{\left( -\frac{{{x}^{6}}}{27} \right)}^{-\frac{1}{3}}}$</div></li><li><div class="list-item-paragraph">${{\left( {{a}^{3}}{{b}^{-9}} \right)}^{-\frac{1}{3}}}$</div></li><li><div class="list-item-paragraph">$\frac{{{c}^{-\frac{1}{3}}}{{d}^{-\frac{1}{4}}}}{{{\left( {{c}^{2}}{{d}^{3}} \right)}^{\frac{1}{3}}}}$</div></li></ol>

Mastering Rational Exponents and Index Laws

<p>Simplify each of the following, expressing your answers in positive index notation.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">${{\left( {{a}^{2}} \right)}^{\frac{5}{6}}}\times {{a}^{\frac{1}{3}}}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">$\sqrt[5]{b}\div \sqrt[3]{{{b}^{2}}}$</div></li></ol></div></div></div>

Simplifying Radical and Rational Powers

<p>Evaluate each of the following without the use of a calculator.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--3" data-layout="3"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">${{27}^{\frac{2}{3}}}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">${{32}^{-\frac{3}{5}}}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="3"><li><div class="list-item-paragraph">${{100}^{1.5}}$</div></li></ol></div></div></div>

Evaluating Fractional and Decimal Indices

<p>Rewrite each of the following in radical form and hence evaluate the results without using a calculator.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{81}^{\frac{1}{4}}}$</div></li><li><div class="list-item-paragraph">${{8}^{-\frac{1}{3}}}$</div></li></ol>
<p>In general, if&nbsp;$m$ and $n$ are positive integers,</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1772950763536-p1jeemmj16d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzI5NTA3NjM1MzYtcDFqZWVtbWoxNmQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcxMjIzNywiZXhwIjoxNzkyNzEyNTM3fQ.eGIx2s_hV8tTiAqxmg3-6noAVafDK0V48mcnUs5B1gw" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>if&nbsp;$a&gt;0$.</p>

Rewrite each of the following in radical form and hence evaluate the results without using a calculator.

In general, if $m$ and $n$ are positive integers,

From Indices to Radicals: Understanding Fractional Powers

<p>Evaluate each of the following without the use of a calculator.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">$\sqrt[4]{16}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">$\sqrt[3]{\frac{27}{125}}$</div></li></ol></div></div></div>

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