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Algebraic Techniques, SM-Bank 094

For the sum \(\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}\), simplify and write an equivalent algebraic fraction.  (3 marks)

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\(\dfrac{13a}{12}\)

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\(\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}\) \(=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{a}{3}\times \dfrac{4}{4}+\dfrac{a}{4}\times \dfrac{3}{3}\)
  \(=\dfrac{6a}{12}+\dfrac{4a}{12}+\dfrac{3a}{12}\)
  \(=\dfrac{13a}{12}\)

Algebraic Techniques, SM-Bank 083

A rectangular prism is known to have a length of  \(5\) cm,  a width of \(3\) cm and a height of \(x+2\) cm. Write an expression for the volume of the prism, giving your answer in simplest form.   (3 marks)

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\((15x+30)\ \text{cm}^3\)

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\(\text{Volume}\) \(=\text{length}\times\text{width}\times\text{height}\)
  \(=5\times 3\times (x+2)\)
  \(=15\times (x+2)\)
  \(=15\times x +15\times 2\)
  \(=(15x+30)\ \text{cm}^3\)

Algebraic Techniques, SM-Bank 076

The side length of a regular hexagon is \(2x+5\). Write an expression for the perimeter of the hexagon.  (2 marks)

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\(6\times(2x+5)\ \ \text{or }\ 6(2x+5)\ \ \text{or }\ 12x+30\)

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\(\text{A regular hexagon has all sides equal}\)

\(\therefore\ \text{Perimeter}\) \(=6\times\text{side length}\)
  \(=6\times(2x+5)\)
  \(=6\times 2x+6\times 5\)
  \(=12x+30\)

Algebraic Techniques, SM-Bank 075

The side length of a square is \(x-3\). Write an expression for the perimeter of the square.  (2 marks)

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\(4\times(x-3)\ \ \text{or }\ 4(x-3)\ \ \text{or }\ 4x-12\)

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\(\text{A square has all sides equal}\)

\(\therefore\ \text{Perimeter}\) \(=4\times\text{side length}\)
  \(=4\times(x-3)\)
  \(=4\times x-4\times 3\)
  \(=4x-12\)

Algebraic Techniques, SM-Bank 074

The side length of an equilateral triangle is \(x+2\). Write an expression for the perimeter of the triangle.  (2 marks)

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\(3\times(x+2)\ \ \text{or }\ 3(x+2)\ \ \text{or }\ 3x+6\)

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\(\text{An equilateral triangle has all sides equal}\)

\(\therefore\ \text{Perimeter}\) \(=3\times\text{side length}\)
  \(=3\times(x+2)\)
  \(=3\times x+3\times 2\)
  \(=3x+6\)

Algebraic Techniques, SM-Bank 065

Simplify the following algebraic expressions, giving your answer in simplest form.

  1. \(\dfrac{9mn}{3}\)  (1 mark)

  2. \(\dfrac{4y}{12x}\)  (1 mark)

  3. \(\dfrac{16a^2}{-4ab}\)  (2 marks)

  4. \(\dfrac{-18b^2}{-24b}\)  (2 marks)

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a.    \(3mn\)

b.    \(\dfrac{y}{3x}\)

c.    \(\dfrac{-4a}{b}\)

d.    \(\dfrac{3b}{4}\)

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a.    \(\dfrac{9mn}{3}\)  \(=\dfrac{3\times 3\times m\times n}{3}\)
    \(=3mn\)

 

b.    \(\dfrac{4y}{12x}\)  \(=\dfrac{4\times y}{4\times 3\times x} \)
    \(=\dfrac{y}{3x}\)

 

c.    \(\dfrac{16a^2}{-4ab}\) \(=\dfrac{4\times 4\times a\times a}{-4\times a\times b}\)
    \(=\dfrac{-4a}{b}\)

 

d.    \(\dfrac{-18b^2}{-24b}\)  \(=\dfrac{-6\times 3\times b\times b}{-6\times 4\times b}\)
    \(=\dfrac{3b}{4}\)

Algebraic Techniques, SM-Bank 060

Simplify the following algebraic expressions, giving your answer in simplest form.

  1. \(-3mn\times 2m\)  (1 mark)

  2. \(4a^2b\times (-3b)\)  (1 mark)

  3. \(-8cd\times (-2d)\)  (1 mark)

  4. \(4a^2b\times 3abc\)  (2 marks)

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a.    \(-6m^2n\)

b.    \(-12a^2b^2\)

c.    \(16cd^2\)

d.    \(12a^3b^2c\)

Show Worked Solution
a.    \(-3mn\times 2m\)  \(=-3\times 2\times m\times n\times m \)
    \(=-6m^2n\)

 

b.    \(4a^2b\times (-3b)\)  \(=4\times -3\times a\times a\times b\times b \)
    \(=-12a^2b^2\)

 

c.    \(-8cd\times (-2d)\) \(=-8\times -2\times c\times d\times d \)
    \(=16cd^2\)

 

d.    \(4a^2b\times 3abc\)  \(=4\times 3\times a\times a\times b\times a\times b\times c \)
    \(=12a^3b^2c\)

Algebraic Techniques, SM-Bank 059

Simplify the following algebraic expressions, giving your answer in simplest form.

  1. \(4pq\times 6r\)  (1 mark)

  2. \(3m\times m\)  (1 mark)

  3. \(7x\times 3x\)  (1 mark)

  4. \(2ab\times 3bc\times c\)  (1 mark)

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a.    \(24pqr\)

b.    \(3m^2\)

c.    \(21x^2\)

d.    \(6ab^2c^2\)

Show Worked Solution
a.    \(4pq\times 6r\)  \(=4\times 6\times p\times q\times r \)
    \(=24pqr\)

 

b.    \(3m\times m\)  \(=3\times m\times m \)
    \(=3m^2\)

 

c.    \(7x\times 3x\) \(=7\times 3\times x\times x \)
    \(=21x^2\)

 

d.    \(2ab\times 3bc\times c\)  \(=2\times 3\times a\times b\times b\times c\times c \)
    \(=6ab^2c^2\)

Algebraic Techniques, SM-Bank 058

Simplify the following algebraic expressions, giving your answer in simplest form.

  1. \(5\times 4y\)  (1 mark)

  2. \(4m\times 3\)  (1 mark)

  3. \(2a\times 3b\times 4\)  (1 mark)

  4. \(4c\times 2b\times 5d\)  (1 mark)

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a.    \(20y\)

b.    \(12m\)

c.    \(24ab\)

d.    \(40bcd\)

Show Worked Solution
a.    \(5\times 4y\)  \(=5\times 4\times y \)
    \(=20y\)

 

b.    \(4m\times 3\)  \(=4\times 3\times m \)
    \(=12m\)

 

c.    \(2a\times 3b\times 4\) \(=2\times 3\times 4\times a\times b \)
    \(=24ab\)

 

d.    \(4c\times 2b\times 5d\)  \(=4\times 2\times 5\times c\times b\times d \)
    \(=40bcd\)

Algebraic Techniques, SM-Bank 057

State the coefficients of \(\large x\) and \(\large y\) that make the expressions below equivalent?  (2 marks)

\(9x+3y-\)
 
 \(x+\)
 
 \(y=6x+8y\)
  
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\(3 , 5\)

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\(9x+3y-\)
 
 \(x+\)
 
 \(y=6x+8y\)

 

\(\text{Equating the }x\ \text{values}\longrightarrow\)  \(9x-\)
 
 \(x=6x\)
  \(\therefore\ \)
 
 \(=3\)

   

\(\text{Equating the }y\ \text{values}\longrightarrow\)  \(3y+\)
 
 \(y=8y\)
  \(\therefore\ \)
 
 \(=5\)

Algebraic Techniques, SM-Bank 055

Write an expression for the perimeter of the rectangle below. Give your answer in simplest form.  (2 marks)

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\(20x+2\)

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\(\text{Method 1}\)

\(\text{Perimeter}\)  \(=7x+1+3x+7x+1+3x\)
  \(=7x+3x+7x+3x+1+1\)
  \(=20x+2\)

 

\(\text{Method 2 (Advanced)}\)

\(\text{Perimeter}\)  \(=2(7x+1)+2\times 3x\)
  \(=14x+2+6x\)
  \(=20x+2\)

Algebraic Techniques, SM-Bank 045 MC

Which of the following is a like term to \(8m\)?

  1. \(3m^2\)
  2. \(8mn\)
  3. \(2m\)
  4. \(4+m\)
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\(C\)

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\(\text{For like terms algebraic parts are identical:}\)

\(\text{Option A:}\longrightarrow m^2\neq m\)

\(\text{Option B:}\longrightarrow mn\neq m\)

\(\text{Option C:}\longrightarrow m = m\ \checkmark\)

\(\text{Option D:}\longrightarrow 4+m\neq m\)

\(\therefore\ 8m\text{ and}\ 2m\text{ are like terms.}\)

\(\Rightarrow C\)

Algebraic Techniques, SM-Bank 044

Evaluate the expression  \(x^2+3x-4\)  when:

  1. \(x=1\)  (2 marks)

  2. \(x=3\)  (2 marks)

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a.    \(0\)

b.    \(14\)

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a.    \(x^2+3x-4\) \(=1^2+3\times 1-4 \)
    \(=1\times 1 +3-4\)
    \(=0\)

 

b.    \(x^2+3x-4\) \(=3^2+3\times 3-4\)
    \(=3\times 3 +3\times 3-4\)
    \(=14\)

Algebraic Techniques, SM-Bank 043

Evaluate the expression  \(-2x^2\)  when:

  1. \(x=1\)  (1 mark)

  2. \(x=-2\)  (2 marks)

  3. \(x=-\dfrac{1}{2}\)  (2 marks)

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a.    \(-2\)

b.    \(-8\)

c.    \(-\dfrac{1}{2}\)

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a.    \(-2x^2\) \(=-2\times 1^2\)
    \(=-2\times 1\times 1\)
    \(=-2\)

 

b.    \(-2x^2\) \(=-2\times (-2)^2\)
    \(=-2\times (-2)\times (-2)\)
    \(=-8\)

 

c.    \(-2x^2\) \(=-2\times \bigg(-\dfrac{1}{2}\bigg)^2\)
    \(=-2\times \bigg(-\dfrac{1}{2}\bigg) \times \bigg(-\dfrac{1}{2}\bigg)\)
    \(=-2\times\dfrac{1}{4}\)
    \(=-\dfrac{1}{2}\)

Algebraic Techniques, SM-Bank 042

Evaluate the expression  \(w^2\)  when:

  1. \(w=4\)  (1 mark)

  2. \(w=-1\)  (2 marks)

  3. \(w=\dfrac{1}{3}\)  (2 marks)

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a.    \(16\)

b.    \(1\)

c.    \(\dfrac{1}{9}\)

Show Worked Solution
a.    \(w^2\) \(=4^2\)
    \(=4\times 4\)
    \(=16\)

 

b.    \(w^2\) \(=(-1)^2\)
    \(=(-1)\times (-1)\)
    \(=1\)

 

c.    \(w^2\) \(=\bigg(\dfrac{1}{3}\bigg)^2\)
    \(=\dfrac{1}{3}\times \dfrac{1}{3}\)
    \(=\dfrac{1}{9}\)