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PSLE Foundation Mathematics

<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

Laws of Indices

IP2U9 HOT Q4

Indices

<p>Simplify $2\div {{\left( \frac{2}{x} \right)}^{-3}}$.</p>

Simplify $2\div {{\left( \frac{2}{x} \right)}^{-3}}$.

Simplifying with Negative Indices and Fractions

Simplifying Expressions with Negative 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.

Rational Indices

Evaluating Indices: Fractions, Roots, and Powers

<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 ${{\left( \frac{2}{n} \right)}^{-3}}$, leaving your answer in positive indices only.</p>

Simplify ${{\left( \frac{2}{n} \right)}^{-3}}$, leaving your answer in positive indices only.

Positive Index Notation with Negative Exponents

<p>Simplify each of the following, giving your answer in positive index notation.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$5{{a}^{4}}\times 3{{a}^{-3}}\div {{a}^{-3}}$</div></li><li><div class="list-item-paragraph">${{\left( 2b \right)}^{0}}\div \left[ 4{{\left( {{b}^{-2}} \right)}^{2}} \right]$</div></li><li><div class="list-item-paragraph">${{\left( -3{{c}^{2}}{{d}^{-2}} \right)}^{2}}\times {{\left( -2{{c}^{-1}}d \right)}^{3}}$</div></li><li><div class="list-item-paragraph">$\frac{e{{f}^{3}}}{4}\div {{\left( \frac{2}{{{e}^{2}}{{f}^{-1}}} \right)}^{-3}}$</div></li><li><div class="list-item-paragraph">$\frac{{{\left( {{g}^{3}}h \right)}^{2}}\times {{g}^{-3}}h}{{{\left( {{g}^{-1}}h \right)}^{-2}}}$</div></li><li><div class="list-item-paragraph">$\frac{15{{p}^{2}}{{q}^{-3}}}{{{\left( 3{{p}^{3}} \right)}^{2}}}\div {{\left( {{p}^{2}}q \right)}^{-1}}$</div></li></ol>

Simplify each of the following, giving your answer in positive index notation.

Consolidation: Expressing Answers in Positive Index Notation

<p>Evaluate each of the following, without the use of a calculator.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( 0.8 \right)}^{-1}}$</div></li><li><div class="list-item-paragraph">${{5}^{-2}}$</div></li><li><div class="list-item-paragraph">${{\left( 3x \right)}^{0}}\div 3{{x}^{-1}}$</div></li><li><div class="list-item-paragraph">$1\div {{\left( 5x \right)}^{-2}}$</div></li><li><div class="list-item-paragraph">${{3}^{0}}-{{3}^{-3}}\times 27$</div></li><li><div class="list-item-paragraph">${{9}^{0}}\times {{\left( \frac{1}{9} \right)}^{-1}}\times {{81}^{-1}}$</div></li></ol>

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

Evaluate Powers with Negative and Zero Indices

<p>Evaluate the following, without the use of a calculator.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{2015}^{0}}$</div></li><li><div class="list-item-paragraph">${{\left( 1\frac{3}{5} \right)}^{0}}$</div></li><li><div class="list-item-paragraph">$-3{{y}^{0}}$</div></li><li><div class="list-item-paragraph">${{\left( 3y \right)}^{0}}$</div></li><li><div class="list-item-paragraph">${{7}^{0}}\times {{7}^{3}}\div {{7}^{2}}$</div></li><li><div class="list-item-paragraph">${{2}^{0}}+{{2}^{3}}$</div></li></ol>

Evaluate the following, without the use of a calculator.

Evaluating Powers of Zero and Index Expressions

<p>Simplify each of the following, giving 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">$18{{a}^{-6}}\div {{\left[ 3\left( {{a}^{-2}} \right) \right]}^{2}}$</div></li></ol><p></p><ol class="i-list" start="3"><li><div class="list-item-paragraph">${{\left( 3{{c}^{2}}{{d}^{-2}} \right)}^{2}}$</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">$5{{b}^{0}}\times 3{{\left( {{b}^{-2}} \right)}^{2}}$</div></li></ol><ol class="i-list" start="4"><li><div class="list-item-paragraph">${{\left( \frac{{{e}^{2}}{{f}^{-1}}}{2} \right)}^{-3}}$</div></li></ol></div></div></div>

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

Simplifying Negative Indices to Positive Form

<p>Evaluate 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">$1\div {{\left( 2x \right)}^{-1}}$</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">$1\div 3{{x}^{-2}}$</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">${{\left( -\frac{3}{x} \right)}^{-4}}$</div></li></ol></div></div></div>

Evaluate the following, without the use of a calculator. 

Negative Indices in Algebraic Expressions

<p>Evaluate 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">${{5}^{-3}}$</div></li></ol><ol class="i-list" start="4"><li><div class="list-item-paragraph">${{\left( \frac{1}{7} \right)}^{-1}}$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p></p><ol class="i-list" start="2"><li><div class="list-item-paragraph">${{\left( -3 \right)}^{-2}}$</div></li></ol><ol class="i-list" start="5"><li><div class="list-item-paragraph">${{2}^{0}}+{{2}^{-3}}$</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">${{\left( \frac{2}{3} \right)}^{-3}}$</div></li></ol><ol class="i-list" start="6"><li><div class="list-item-paragraph">${{\left( {{2}^{-2}} \right)}^{3}}\times {{\left( {{2}^{-3}} \right)}^{-2}}$</div></li></ol></div></div></div>
<p style="margin-top: 1.5rem"><strong>Important:</strong><br><em>From Worked Example 1 <strong>(iii)</strong> and <strong>(iv)</strong>, we can have the following result:</em></p>
<p style="text-align: center">${{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}$</p>

**Important:**
*From Worked Example 1 **(iii)** and **(iv)**, we can have the following result:*

${{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}$

Negative Indices

<p>Evaluate 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">${{5}^{3}}\times {{5}^{0}}\div {{5}^{2}}$</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">${{5}^{0}}-{{5}^{2}}$</div></li></ol></div></div></div>
<p><em>Tip: Remember laws of indices does not apply to addition and subtraction!</em></p>

*Tip: Remember laws of indices does not apply to addition and subtraction!*

Zero Indices and Common Mistakes

<p>Evaluate 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;"><p></p><ol class="i-list"><li><div class="list-item-paragraph">${{13}^{0}}$</div></li></ol><ol class="i-list" start="3"><li><div class="list-item-paragraph">${{\left( 5a \right)}^{0}}$</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">${{\left( -\frac{3}{7} \right)}^{0}}$</div></li></ol><ol class="i-list" start="4"><li><div class="list-item-paragraph">$5{{a}^{0}}$</div></li></ol></div></div></div>

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