A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that&nbsp;$3x^2 - 5x + 21$ is always positive for all real values of $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The curve $y = ax^n$, where&nbsp;$a$ and $n$ are constants, passes through $(2, 48)$, $(3, 108)$ and $(k, 192)$. Find the values of $a$, $n$ and $k$ where&nbsp;$k &gt; 0$.<span class="inline-marks">[3]</span></div></li></ol>

Equality of Indices

Quadratic Expressions Positive or Negative for All Real Values

2024 CCHY P1 Q3

Quadratic Functions, Equations and Inequalities

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>

Zero and Negative Indices

Laws of Indices

IP2U9 HOT Q4

Indices

<p>Given that $2^{2x+3} \times 5^{x-2} = 8^x \times 5^{2x}$, evaluate $10^x$.</p>

Given that $2^{2x+3} \times 5^{x-2} = 8^x \times 5^{2x}$, evaluate $10^x$.

<p>Given that $3^{x+2} \times 7^{x-1} = 42^{2x} \div 2^{2x}$, evaluate $21^x$.</p>

Given that $3^{x+2} \times 7^{x-1} = 42^{2x} \div 2^{2x}$, evaluate $21^x$.

<p>Given that $\frac{3^x}{2^{x+1}} = \frac{5}{4^x}$, find the value of $216^x$.</p>

Given that $\frac{3^x}{2^{x+1}} = \frac{5}{4^x}$, find the value of $216^x$.

<p>Given that $5^{x+3} \times 4^{x-2} = 2^{x+5}$, find the value of $10^x$.</p>

Given that $5^{x+3} \times 4^{x-2} = 2^{x+5}$, find the value of $10^x$.

<p>Without using a calculator, solve $\frac{7^x}{27} = 567^x$.</p>

Without using a calculator, solve $\frac{7^x}{27} = 567^x$.

<p>Evaluate $6^x$, given that $3^{2x+3}\times2^{x+5}=3^{x+4}$.</p>

Evaluate $6^x$, given that $3^{2x+3}\times2^{x+5}=3^{x+4}$.

<p>Solve the following exponential equations by using index laws to express both sides with the same base.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Solve $2^x = 32$</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve $5^x \times 3^{2x-1} = \frac{1}{135}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve $5^{3x-1} = 625$</span></div></li></ol>

Solve the following exponential equations by using index laws to express both sides with the same base.

Solve $5^x \times 3^{2x-1} = \frac{1}{135}$

<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>Assume $a&gt;0$ and $a\ne1$.</p>
<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>Assume $x&gt;0$ and $x\ne1$.</p>
<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 $p$.

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>Assume $a&gt;0$ and $a\ne1$.</p>
<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>Find the integer $n$ such that&nbsp;${{8}^{n}}+{{8}^{n}}+{{8}^{n}}+{{8}^{n}}={{2}^{50}}$.<br><br></p>

Find the integer $n$ such that ${{8}^{n}}+{{8}^{n}}+{{8}^{n}}+{{8}^{n}}={{2}^{50}}$.



Challenging Question involving Indices

<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;${{7}^{-2}}\times {{7}^{x}}=1$, find the value of $x$.</div></li><li><div class="list-item-paragraph">Solve ${{7}^{x-1}}={{\left( \frac{1}{49} \right)}^{\frac{3}{2}}}$ .</div></li></ol>

Math Question Bank