smc-4694-30-Simplify

Algebraic Techniques, SM-Bank 086

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

\(\dfrac{5xy\times 3xy}{10\times xy}\)

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\(\dfrac{3xy}{2}\)

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\(\dfrac{5xy\times 3xy}{10\times xy}\)	\(=\dfrac{15\times xy\times xy}{10\times xy}\)
	\(=\dfrac{15xy}{10}\)
	\(=\dfrac{3xy}{2}\)

Algebraic Techniques, SM-Bank 085

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

\(\dfrac{3\times ab\times 4a}{a\times 2b}\)

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\(6a\)

Show Worked Solution

\(\dfrac{3\times ab\times 4a}{a\times 2b}\)	\(=\dfrac{12\times a\times ab}{2\times ab}\)
	\(=6a\)

Algebraic Techniques, SM-Bank 084

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

\(\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}\)

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\(n\)

Show Worked Solution

\(\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}\)	\(=\dfrac{60\times m\times mn\times n}{60\times m\times mn}\)
	\(=n\)

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 082

Write a simplified expression for the volume of the rectangular prism below. (3 marks)

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\(24abc\)

Show Worked Solution

\(\text{Volume}\)	\(=\text{length}\times\text{width}\times\text{height}\)
	\(=4c\times 3a\times 2b\)
	\(=24abc\)

Algebraic Techniques, SM-Bank 081

Write a simplified expression for the volume of the cube below. (3 marks)

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\(8x^3\)

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\(\text{Volume}\)	\(=\text{(side)}^3\)
	\(=(2x)^3\)
	\(=2x\times 2x\times 2x\)
	\(=8x^3\)

Algebraic Techniques, SM-Bank 080

Write a simplified expression for the area of the shape below. (3 marks)

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\(48ab\)

Show Worked Solution

\(\text{Area}\)	\(=\text{Area big rectangle}-\text{Area small rectangle}\)
	\(=6a\times 10b-2a\times 6b\)
	\(=60ab-12ab\)
	\(=48ab\)

Algebraic Techniques, SM-Bank 079

Write a simplified expression for the area of the rectangle below. (2 marks)

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\(28b^2\)

Show Worked Solution

\(\text{Area}\)	\(=\text{length}\times \text{width}\)
	\(=7b\times 4b\)
	\(=28b^2\)

Algebraic Techniques, SM-Bank 078

Write a simplified expression for the area of the rectangle below. (2 marks)

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\(10mn\)

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\(\text{Area}\)	\(=\text{length}\times \text{width}\)
	\(=5m\times 2n\)
	\(=10mn\)

Algebraic Techniques, SM-Bank 077

Write a simplified expression for the area of the triangle below. (2 marks)

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\(3x\)

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\(\text{Area}\)	\(=\dfrac{1}{2}\times\text{base}\times \text{height}\)
	\(=\dfrac{1}{2}\times 2x\times 3\)
	\(=\dfrac{1}{2}\times 6x\)
	\(=3x\)

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

Show Worked Solution

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

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

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 073

The perimeter of a regular hexagon is equal to \(4x+3y\). Write an expression for the side length of the hexagon. (2 marks)

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\(\dfrac{4x+3y}{6}\)

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

\(\therefore\ \text{Side length}\)	\(=\dfrac{\text{Perimeter}}{6}\)
	\(=\dfrac{4x+3y}{6}\)

Algebraic Techniques, SM-Bank 072

The perimeter of a square is equal to \(5q-3\). Write an expression for the side length of the square. (2 marks)

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\(\dfrac{5q-3}{4}\)

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

\(\therefore\ \text{Side length}\)	\(=\dfrac{\text{Perimeter}}{4}\)
	\(=\dfrac{5q-3}{4}\)

Algebraic Techniques, SM-Bank 071

The perimeter of an equilateral triangle is equal to \(4m+5\). Write an expression for the side length of the triangle. (2 marks)

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\(\dfrac{4m+5}{3}\)

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

\(\therefore\ \text{Side length}\)	\(=\dfrac{\text{Perimeter}}{3}\)
	\(=\dfrac{4m+5}{3}\)

Algebraic Techniques, SM-Bank 070 MC

Which of the following expressions is equivalent to \((m\times 4)÷(3\times n)\)?

\(\dfrac{3m}{4n}\)
\(\dfrac{3n}{4m}\)
\(\dfrac{3mn}{4}\)
\(\dfrac{4m}{3n}\)

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\(D\)

Show Worked Solution

\((m\times 4)÷(3\times n)\)

\(=\dfrac{m\times 4}{3\times n}\)

\(=\dfrac{4m}{3n}\)

\(\Rightarrow D\)

Algebraic Techniques, SM-Bank 069 MC

Which of the following expressions is equivalent to \(m÷(3\times n)\)?

\(\dfrac{m}{3n}\)
\(\dfrac{3}{mn}\)
\(\dfrac{3m}{n}\)
\(\dfrac{mn}{3}\)

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\(A\)

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\(m÷(3\times n)\)

\(=\dfrac{m}{3\times n}\)

\(=\dfrac{m}{3n}\)

\(\Rightarrow A\)

Algebraic Techniques, SM-Bank 068 MC

Which of the following expressions is equivalent to \(x÷2\times y\)?

\(\dfrac{2x}{y}\)
\(\dfrac{xy}{2}\)
\(\dfrac{2y}{x}\)
\(\dfrac{x}{2y}\)

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\(B\)

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\(x÷2\times y\)

\(=\dfrac{x}{2}\times y\)

\(=\dfrac{xy}{2}\)

\(\Rightarrow B\)

Algebraic Techniques, SM-Bank 067 MC

Which of the following expressions is equivalent to \(a÷b\times c\)?

\(\dfrac{ac}{b}\)
\(\dfrac{a}{bc}\)
\(\dfrac{ab}{c}\)
\(\dfrac{bc}{a}\)

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\(A\)

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\(a÷b\times c\)

\(=\dfrac{a}{b}\times c\)

\(=\dfrac{ac}{b}\)

\(\Rightarrow A\)

Algebraic Techniques, SM-Bank 066 MC

Which of the following expressions is equivalent to \(a÷(b\times c)\)?

\(\dfrac{ac}{b}\)
\(\dfrac{a}{bc}\)
\(\dfrac{ab}{c}\)
\(\dfrac{bc}{a}\)

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\(B\)

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\(a÷(b\times c)\)

\(=\dfrac{a}{b\times c}\)

\(=\dfrac{a}{bc}\)

\(\Rightarrow B\)

Algebraic Techniques, SM-Bank 065

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

\(\dfrac{9mn}{3}\) (1 mark)

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\(\dfrac{4y}{12x}\) (1 mark)

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

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\(\dfrac{-18b^2}{-24b}\) (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 064 MC

Which of the following algebraic expressions is equivalent to \(\dfrac{20xy}{5yz}\)?

\(5xy^2z\)
\(4xy^2z\)
\(\dfrac{4}{z}\)
\(\dfrac{4x}{z}\)

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\(D\)

Show Worked Solution

\(\dfrac{20xy}{5yz}\)

\(=\dfrac{5\times 4\times x\times y}{5\times y\times z}\)

\(=\dfrac{4x}{z}\)

\(\Rightarrow D\)

Algebraic Techniques, SM-Bank 063 MC

Which of the following algebraic expressions is equivalent to \(x\times x\times y\times x\times y\)?

\(6xy\)
\(x^2y^3\)
\(x^3y^2\)
\(3x+2y\)

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\(C\)

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\(x\times x\times y\times x\times y\)

\(=x\times x\times x\times y\times y\)

\(=x^3y^2\)

\(\Rightarrow C\)

Algebraic Techniques, SM-Bank 062 MC

Which of the following algebraic expressions is equivalent to \(3a\times 2b\times ab\times 2\)?

\(7ab\)
\(12ab\)
\(6a^2b^2\)
\(12a^2b^2\)

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\(D\)

Show Worked Solution

\(3a\times 2b\times ab\times 2\)

\(=3\times 2\times 2\times a\times a\times b\times b\)

\(=12a^2b^2\)

\(\Rightarrow D\)

Algebraic Techniques, SM-Bank 061 MC

Which of the following algebraic expressions is equivalent to \(4\times x\times y\times 2\)?

\(42xy\)
\(8xy\)
\(4xy^2\)
\(16xy\)

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\(B\)

Show Worked Solution

\(4\times x\times y\times 2\)

\(=4\times 2\times x\times y\)

\(=8xy\)

\(\Rightarrow B\)

Algebraic Techniques, SM-Bank 060

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

\(-3mn\times 2m\) (1 mark)

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\(4a^2b\times (-3b)\) (1 mark)

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\(-8cd\times (-2d)\) (1 mark)

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\(4a^2b\times 3abc\) (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.

\(4pq\times 6r\) (1 mark)

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\(3m\times m\) (1 mark)

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\(7x\times 3x\) (1 mark)

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\(2ab\times 3bc\times c\) (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.

\(5\times 4y\) (1 mark)

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\(4m\times 3\) (1 mark)

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

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