Make the following numbers using FOUR 1s and any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$, $-$, $\times$, $\div$, $($ $)$, etc by the Sign of One. Click the numbers below to see the answers. The first one is done for you.

# Algebra

# Number Curiosity

Are these number beautiful? \begin{align} \displaystyle 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 &= 98 \\ 123 \times 8 + 3 &= 987 \\ 1234 \times 8 + 4 &= 9876 \\ 12345 \times 8 + 5 &= 98765 \\ 123456 \times 8 + 6 &= 987654 \\ 1234567 […]

# Nicomachus Theorem

$\textbf{Nicomachus}$ discovered “Nicomachus Theorem” interesting number patterns involving cubes and sums of odd numbers. Nicommachus was born in Roman Syria (now, Jerash, Jordan) around 100 AD. He wrote in Greek was a Pythagorean. $\textbf{Nicomachus Theorem: Cubes and Sums of Odd numbers}$ \begin{eqnarray*} 1 &=& 1^3 \\ 3 + 5 &=& 8 = 2^3 \\ 7 […]

# 10 Deadly Common Algebra Mistakes

Share0 Share +10 Tweet0 Common Patterns of Algebra Mistakes Students often make Common Algebra Mistakes due to confusions such as expand and simplify rules, fractions, indices and equations which that lead the students to the wrong answer. Also these error patterns are very basic and quite easily rectified. Check yourself whether you make similar mistakes, […]

# 4 important types of absolute value equation

Share0 Share +10 Tweet0 Absolute Value Equation There are 4 main type of absolute value equation regarding whether there are; absolute value and a static value absolute value and an expression involving unknown pronumerals two absolute values and a value two absolute values in both sides Type 1 of Absolute Value Equation Solve \( | […]

# Harder Factorise

Share0 Share +10 Tweet0 Algebra – Harder Factorise or Factoring can be done by collecting common factors and using sums and products of factors. $$x^2 – (a+b)x + ab = (x-a)(x-b)$$ Question 1 Factorise \( (x^2-3x)^2 -2x^2 + 6x – 8 \). Question 2 Factorise \( (x-1)(x-3)(x+2)(x+4) + 24 \). Question 3 Factorise \( x^4 […]

# Solving Radical Equations

Share0 Share +10 Tweet0 Solving radical equations are required to isolate the radicals or surds to one side of the equations. Then square both sides. Here we have two important checkpoints. Checkpoint 1 for Solving radical equations Make sure the whole both sides are to be squared, but not squaring individual terms. This is what […]

# Inequality with Variables in the Denominator

Share0 Share +10 Tweet0 Solving Inequality with Variables in the Denominator requires special cares due to the direction of the inequalities. Let’s have a look at the following key points. Key Point 1 \( \begin{aligned} \displaystyle \require{color} \frac{1}{x} &\ge 2 \\ \frac{1}{x} \times x &\ge 2 \times x &\color{green} \text{Many of you may think this […]

# Inequality Proofs

Share0 Share +10 Tweet0 Inequality Proofs can be done many ways. For proving \(A \ge B \), one of the easiest ways is to show \(A – B \ge 0 \). Worked Example of Inequality Proofs If \(a, b\) and \(c\) are positive real numbers and \(a+b\ge c \), prove that \( \displaystyle \frac{a}{1+a} + […]

# Rationalising Denominators

Share0 Share +10 Tweet0 Rationalising Denominators simplifies the fractions by multiplying the conjugates of the denominators. Worked Example of Rationalising Denominators Simplify \( \displaystyle\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \).

# Absolute Value Inequalities

Share0 Share +10 Tweet0 Absolute Value Inequalities are usually proved by the absolute value of a certain value is greater than or equal to it. The square of the value is equal to the square of its absolute value. Proof of Absolute Value Inequalities Prove \(|a| + |b| \ge |a+b|\).

# Inequalities using Arithmetic Mean Geometric Mean

Share0 Share +10 Tweet0 Arithmetic Mean of \(a\) and \(b\) is always greater than or equal to the Geometric Mean of \(a\) and \(b\), for all positive real numbers with with equality if and only if \(a = b\). This is also called AM-GM (Arithmetic Mean Geometric Mean) inequality. \(\require{color}\) $$ \begin{aligned} \frac{a + b}{2} […]

# 3 Ways of Evaluating Nested Square Roots

Share0 Share +10 Tweet0 Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to understand how the students can handle “…”. This enables us to setup a quadratic equation to evaluate its exact value using the quadratic formula, \(\require{color}\) $$x= \frac{-b \ \pm \sqrt{b^2 – […]