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- Use prime factors to explain why 18 x 63 is not a perfect square. - The number 18p is a perfect cube. Find the smallest positive integer value of p.

Use prime factors to explain why $18\times63$ is not a perfect square.

The number $18p$ is a perfect cube. Find the smallest positive integer value of $p$.

Prime Factorisation and Its Applications

Laws of Indices

2025 SST PRELIMS P1 Q2

Indices

Primes, HCF and LCM

Sec 3/4 E Math T03-Q03 - Solving an exponential equation

<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$x={{16}^{18}}$ and $y={{18}^{16}}$, without using a calculator, determine whether $x&gt;y$,  showing your working clearly.</div></li><li><div class="list-item-paragraph">Arrange ${{3}^{312}}$, ${{6}^{234}}$ and ${{13}^{156}}$ in descending order.</div></li><li><div class="list-item-paragraph">Given that&nbsp;${{\left( {{x}^{2}}-x-1 \right)}^{x+2000}}=1$, find all the integer solutions.</div></li></ol>

Equality of Indices

Zero and Negative Indices

IP2U9 HOT Q4

<ol class="abc-list"><li data-list-item-id="item-1787979660079-oowocqkat" data-list-item-path="item-1787979660079-oowocqkat"><div class="list-item-paragraph"><p>Simplify $\frac{4^{x+2}-2^{2x+1}}{8^x(4^{1-x})}$ and express it in the form $a(2^{x-b})$ where $a$ and $b$ are integers.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1787979661076-m9xdoxl1k" data-list-item-path="item-1787979661076-m9xdoxl1k"><div class="list-item-paragraph"><p>Hence, solve for $x$ if $\frac{4^{x+2}-2^{2x+1}}{8^x(4^{1-x})} = 15$.</p><span class="inline-marks">[2]</span></div></li></ol>

Solving Exponential and Power Equations Using Logarithms

Exponential and Logarithmic Functions

<p>Given that $p=16^x$ and $q=2^y$, express $\frac{2p}{q^3}$ as a power of $2$.</p>

Given that $p=16^x$ and $q=2^y$, express $\frac{2p}{q^3}$ as a power of $2$.

<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>

Rational Indices

Simplifying Expressions Using Advanced Index Laws

<p>If&nbsp;${{7}^{m}}=3$ and ${{7}^{n}}=4$, find the value of ${{7}^{2m+3n}}$.</p>

If ${{7}^{m}}=3$ and ${{7}^{n}}=4$, find the value of ${{7}^{2m+3n}}$.

Challenging Question involving Indices

<p>Simplify ${{\left( {{5}^{p}} \right)}^{3}}\times {{\left( {{5}^{2}} \right)}^{p}}\div {{5}^{5p}}$.</p>

Simplify ${{\left( {{5}^{p}} \right)}^{3}}\times {{\left( {{5}^{2}} \right)}^{p}}\div {{5}^{5p}}$.

Laws of Indices with Variable Powers

<p>Simplify $\frac{8{{x}^{8}}{{y}^{4}}}{{{\left( 2x{{y}^{2}} \right)}^{2}}}\times \frac{{{\left( 4{{x}^{2}}{{y}^{2}} \right)}^{2}}}{{{\left( 3xy \right)}^{2}}}$.</p>

Simplify $\frac{8{{x}^{8}}{{y}^{4}}}{{{\left( 2x{{y}^{2}} \right)}^{2}}}\times \frac{{{\left( 4{{x}^{2}}{{y}^{2}} \right)}^{2}}}{{{\left( 3xy \right)}^{2}}}$.

Advanced Laws of Indices: Fractions and Powers

<p>Simplify each of the following</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{a}^{7}}\times {{a}^{3}}\div {{\left( {{a}^{3}} \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( 2b \right)}^{5}}\div 8{{b}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{3{{x}^{2}}}{{{x}^{3}}} \right)}^{3}}\div \frac{27{{x}^{7}}}{{{x}^{21}}}$</div></li><li><div class="list-item-paragraph">${{\left( {{c}^{2}}d \right)}^{4}}\times {{\left( {{c}^{4}}{{d}^{3}} \right)}^{5}}$</div></li><li><div class="list-item-paragraph">$5{{\left( ef \right)}^{3}}\times 10e{{f}^{2}}$</div></li><li><div class="list-item-paragraph">$16{{m}^{8}}{{n}^{7}}\div {{\left( -2{{m}^{3}}{{n}^{2}} \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{{{p}^{2}}}{q} \right)}^{6}}\times {{\left( \frac{2{{q}^{2}}}{-3{{p}^{5}}} \right)}^{3}}$</div></li></ol>

Advanced Simplification with Index Laws

<p>Simplify each of the following</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{a}^{7}}\div {{a}^{3}}$</div></li><li><div class="list-item-paragraph">$8{{b}^{5}}\div 2{{b}^{2}}$</div></li><li><div class="list-item-paragraph">$7{{c}^{2}}{{d}^{3}}\div 14c{{d}^{2}}$</div></li><li><div class="list-item-paragraph">$3{{e}^{5}}{{f}^{2}}\div \left( -9{{e}^{3}}f \right)$</div></li></ol>

Simplifying Algebraic Expressions Using Division of Indices

<p>Simplify each of the following</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{a}^{5}}\times {{a}^{3}}\times a$</div></li><li><div class="list-item-paragraph">$3{{b}^{4}}\times 2{{b}^{3}}$</div></li><li><div class="list-item-paragraph">$5{{c}^{5}}{{d}^{3}}\times 4c{{d}^{2}}$</div></li><li><div class="list-item-paragraph">$-2{{e}^{2}}f\times 7{{e}^{3}}{{f}^{2}}$</div></li></ol>

Simplifying Algebraic Expressions Using Index Laws

<p>Simplify each of the following</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">${{\left( \frac{3}{{{a}^{2}}} \right)}^{3}}$</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">${{\left( \frac{{{b}^{2}}}{c} \right)}^{3}}\times \frac{{{c}^{7}}}{{{b}^{2}}}$</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">$\frac{6{{d}^{3}}}{e}\div {{\left( \frac{-3d}{{{e}^{2}}} \right)}^{2}}$</div></li></ol></div></div></div>

Applying Law of Indices

<p>Simplify each of the following</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">${{\left( 3{{x}^{2}} \right)}^{2}}\times {{\left( 2{{x}^{5}} \right)}^{3}}$</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">$16{{x}^{8}}{{y}^{3}}\div {{\left( -2{{x}^{2}}y \right)}^{3}}$</div></li></ol></div></div></div>

Applying Law 1, 2, 3 and 4 of Indices

<p>Simplify each of the following</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">${{\left( 2{{x}^{3}} \right)}^{5}}$</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">${{\left( -3x{{y}^{2}} \right)}^{3}}$</div></li></ol></div></div></div>

Applying Law 3 and 4 of Indices

<p>Simplify each of the following</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">${{7}^{10}}\div {{7}^{3}}$</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">$\left( 6{{y}^{6}} \right)\div \left( 2{{y}^{3}} \right)$</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">$\left( -36{{a}^{7}}{{b}^{4}} \right)\div \left( -3{{a}^{2}}b \right)$</div></li></ol></div></div></div>

Applying Law 2 of Indices

<p>Simplify each of the following</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">${{x}^{2}}\times {{x}^{3}}$</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">$\left( 3{{y}^{2}} \right)\times \left( 8{{y}^{7}} \right)$</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">$\left( 2a{{b}^{3}} \right)\times \left( -3{{a}^{2}}{{b}^{4}} \right)$</div></li></ol></div></div></div>

Applying Law 1 of Indices

<p>Given that&nbsp;$98.23^3\approx947834$ to 6 significant figures, use this result, without a calculator, to estimate $0.009823^3$ to 6 significant figures.</p>

Given that $98.23^3\approx947834$ to 6 significant figures, use this result, without a calculator, to estimate $0.009823^3$ to 6 significant figures.

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