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<p>If&nbsp;$\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.</p>

If $\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.

Equality of Indices

Index Notation

Indices

Laws of Indices

Exponential and Logarithmic Functions

<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>

IP2U9 HOT Q4

<p>Simplify the expression $\sqrt{\sqrt{3}-\sqrt{3-\sqrt{3}}} \times \sqrt{\sqrt{3}+\sqrt{3-\sqrt{3}}}$.</p>

Simplify the expression $\sqrt{\sqrt{3}-\sqrt{3-\sqrt{3}}} \times \sqrt{\sqrt{3}+\sqrt{3-\sqrt{3}}}$.

IP2U9 HOT Q3

<p>Given that&nbsp;$63^x = 35^y = 2205$, find the value of $\frac{x + 4xy + y}{2xy - x - y}$.</p>

Given that $63^x = 35^y = 2205$, find the value of $\frac{x + 4xy + y}{2xy - x - y}$.

IP2U9 HOT Q2

<p>Given that&nbsp;$5^x = 9^y = 2025$, find the value of $\frac{xy}{x + y}$.</p>

Given that $5^x = 9^y = 2025$, find the value of $\frac{xy}{x + y}$.

IP2U9 HOT Q1

<p>Given that&nbsp;$\frac{8\times {{2}^{x}}}{\sqrt{{{2}^{x}}}}=32$ , find the value of $x$.</p>

Given that $\frac{8\times {{2}^{x}}}{\sqrt{{{2}^{x}}}}=32$ , find the value of $x$.

Laws of Indices – Exponential Equation

<p>Given that&nbsp;${{2}^{x}}\times 7=56$, find the value of $x$.</p>

Given that ${{2}^{x}}\times 7=56$, find the value of $x$.

Use Factorisation to Solve for Powers

<p>Given that ${{3}^{27}}\div \frac{1}{9}={{3}^{k}}$, find $k$.</p>

Given that ${{3}^{27}}\div \frac{1}{9}={{3}^{k}}$, find $k$.

Laws of Indices: Division and Negative Powers

<p>Solve</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{3}^{2x+1}}=1$</div></li><li><div class="list-item-paragraph">$\sqrt[3]{{{8}^{x}}}=0.125$</div></li></ol>

Solving Exponential Equations Involving Powers and Roots

<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>

Simplifying Expressions Using Advanced Index Laws

<p>Given that&nbsp;${{a}^{5}}\times {{a}^{2}}\div \sqrt[3]{a}={{a}^{k}}$, find the value of $k$.</p>

Given that ${{a}^{5}}\times {{a}^{2}}\div \sqrt[3]{a}={{a}^{k}}$, find the value of $k$.

Laws of Indices: Solve for the Unknown Exponent

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

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

Simplifying with Negative Indices and Fractions

Simplifying Expressions with Negative Indices

<p>Given that&nbsp;$\frac{x\times {{x}^{3}}}{{{x}^{p}}}=\sqrt{x}$ , find the value of $p$.</p>

Given that $\frac{x\times {{x}^{3}}}{{{x}^{p}}}=\sqrt{x}$ , find the value of $p$.

Rational Indices

Laws of Indices: Solve for the Unknown Power

<p>Given that&nbsp;${{2}^{27}}\div 32={{64}^{x}}$, find the value of $x$.</p>

Given that ${{2}^{27}}\div 32={{64}^{x}}$, find the value of $x$.

Solving Index Equations by Matching Bases

<p>If&nbsp;${{\left( {{a}^{x}} \right)}^{\frac{2}{3}}}=\frac{1}{{{a}^{2}}}$, find the value of $x$.</p>

If ${{\left( {{a}^{x}} \right)}^{\frac{2}{3}}}=\frac{1}{{{a}^{2}}}$, find the value of $x$.

Solving Exponent Equations with Fractional Powers

<p>Solve</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{3}^{x}}=81$</div></li><li><div class="list-item-paragraph">${{\left( \frac{1}{8} \right)}^{x}}=16$</div></li><li><div class="list-item-paragraph">${{10}^{-x}}=0.001$</div></li><li><div class="list-item-paragraph">${{3}^{2x-1}}=243$</div></li><li><div class="list-item-paragraph">${{2}^{x+3}}\times {{4}^{x}}={{8}^{3x-2}}$</div></li><li><div class="list-item-paragraph">${{5}^{14}}\div \frac{1}{25}={{5}^{x}}$</div></li></ol>

Solving Exponential Equations Using Index Laws

<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.

Math Question Bank