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Indices, SM-Bank 058

  1. Write 216 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt[3]{216}\).  (2 marks)

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a.    \(2\times 2\times 2\times 3\times 3\times 3\)

b.    \(6\)

Show Worked Solution
a.    \(216\) \(=2\times 108\)
    \(=2\times 2\times 54\)
    \(=2\times 2\times 2\times 27\)
    \(=2\times 2\times 2\times 3\times 9\)
    \(=2\times 2\times 2\times 3\times 3\times 3\)

 

b.    \(\sqrt[3]{216}\) \(=\sqrt[3]{2\times 2\times 2\times 3\times 3\times 3}\)
    \(=\sqrt[3]{(2\times 3)\times (2\times 3)\times (2\times 3)}\)
    \(=\sqrt[3]{6\times 6\times 6}\)
    \(=6\)

Indices, SM-Bank 057

  1. Write 8 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt[3]{8}\).  (1 mark)

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a.    \(2\times 2\times 2\)

b.    \(2\)

Show Worked Solution
a.    \(8\) \(=2\times 4\)
    \(=2\times 2\times 2\)

 

b.    \(\sqrt[3]{8}\) \(=\sqrt[3]{2\times 2\times 2}\)
    \(=2\)

Indices, SM-Bank 056

Using divisibility tests, find the largest number less than 500 that is divisible by 9.   (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{A number is divisible by }9\ \text{if the sum of the digits is divisible by }9.\)

\(\text{Checking numbers less than } 500\)

\(499=4+9+9=22\ \Rightarrow\ \text{ not divisible by 9}\)

\(498=4+9+8=21\ \Rightarrow\ \text{ not divisible by 9}\)

\(497=4+9+7=20\ \Rightarrow\ \text{ not divisible by 9}\)

\(496=4+9+6=19\ \Rightarrow\ \text{ not divisible by 9}\)

\(495=4+9+5=18\ \Rightarrow\ \text{ divisible by 9}\ \checkmark\)

\(\therefore\ 495\ \text{is the largest number less than }500\ \text{that is divisible by }9.\)

Indices, SM-Bank 055

Using divisibility tests, find the smallest number greater than 200 that is divisible by 6.   (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{A number is divisible by }6\ \text{if it is divisible by both }2\ \text{and }3.\)

\(\text{A number is divisible by }2\ \text{if the last digit is  }2 , 4 , 6 , 8 , 0.\)

\(\therefore\ \text{The possible numbers are in the pattern }202 , 204 , 206 …\ \text{as they are all divisible by }2\)

\(\text{A number is divisible by }3\ \text{if the sum of the digits is divisible by }3.\)

\(2+0+2=4\ \Rightarrow\ \text{ not divisible by 3}\)

\(2+0+4=6\ \Rightarrow\ \text{ divisible by 3}\ \checkmark\)

\(\therefore\ 204\ \text{is the smallest number greater than }200\ \text{that is divisible by }6.\)

Indices, SM-Bank 054

Using divisibility tests, find the smallest number greater than 1000 that is divisible by 6.   (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{A number is divisible by }6\ \text{if it is divisible by both }2\ \text{and }3.\)

\(\text{A number is divisible by }2\ \text{if the last digit is  }2 , 4 , 6 , 8 , 0.\)

\(\therefore\ \text{The first possible number is }1002\ \text{as it is divisible by }2\)

\(\text{A number is divisible by }3\ \text{if the sum of the digits is divisible by }3.\)

\(1+0+0+2=3\ \Rightarrow\ \text{divisible by 3}\ \checkmark\)

\(\therefore 1002\ \text{is the smallest number greater than }1000\ \text{that is divisible by }6.\)

Indices, SM-Bank 053

A number is divisible by 12 if it is also divisible by 3 and 4.

Prove, using the divisibility tests for 3 and 4, that 756 is divisible by 12.   (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{A number is divisible by }3\ \text{if the sum of the digits is divisible by }3.\)

\(7+5+6=18\ \Rightarrow\ \text{divisible by 3}\ \checkmark\)

\(\text{For divisibility by }4\ \text{the last 2 digits in the number must be divisible by }4. \)

\(56\ ÷\ 4=14\ \Rightarrow\ \text{divisible by 4}\ \checkmark\)

\(\therefore 756\ \text{is divisible by }12\ \text{as it is divisible by both }3\ \text{and }4.\)

Indices, SM-Bank 052

Prove using divisibility tests that 282 is divisible by 6.   (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{A number is divisible by }6\ \text{if it is divisible by both }2\ \text{and }3.\)

\(\text{For divisibility by }2\ \text{the number must end in }2 , 4 , 6 , 8 , 0.\ \checkmark\)

\(\text{For divisibility by }3\ \text{the sum of the digits must be divisible by }3.\)

\(2+8+2=12\ \Rightarrow\ \text{divisible by 3}\ \checkmark\)

\(\therefore 282\ \text{is divisible by }6\)

Indices, SM-Bank 051 MC

Which of the following numbers is not divisible by \(9\)?

  1. 234
  2. 1845
  3. 506
  4. 126
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\(C\)

Show Worked Solution

\(\text{A number is divisible by }9\ \text{if the sum of the digits is divisible by }9.\)

\(\text{Option A:}\rightarrow\) \(\ 2+3+4=9\ \checkmark\)
\(\text{Option B:}\rightarrow\) \(\ 1+8+4+5=18\ \checkmark\)
\(\text{Option C:}\rightarrow\) \(\ 5+0+6=11\ -\text{not divisible by }9\)
\(\text{Option D:}\rightarrow\) \(\ 1+2+6=9\ \checkmark\)

 
\(\Rightarrow C\)

Indices, SM-Bank 050 MC

Which of the following numbers is not divisible by \(5\)?

  1. \(1800\)
  2. \(95\)
  3. \(102\)
  4. \(1\ 202\ 005\)
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\(C\)

Show Worked Solution

\(\text{A number is divisible by }5\ \text{if the last digit is a }5 \text{ or }0.\)

\(\text{Option C:}\rightarrow\) \(\ 102\ \text{does not end in }5 \text{ or }0.\)

 
\(\Rightarrow C\)

Indices, SM-Bank 049 MC

Which of the following numbers is not divisible by 2?

  1. 505
  2. 44
  3. 1258
  4. 1202
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\(A\)

Show Worked Solution

\(\text{A number is divisible by }2\ \text{if the last digit is a }2 , 4 , 6 , 8 \text{ or }0.\)

\(\text{Option A:}\rightarrow\) \(505\ \text{does not end in }2 , 4 , 6 , 8 \text{ or }0.\)

 
\(\Rightarrow A\)

Indices, SM-Bank 048 MC

Which of the following numbers is not divisible by 4?

  1. 112
  2. 32
  3. 502
  4. 608
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\(C\)

Show Worked Solution

\(\text{A number is divisible by }4\ \text{if the last 2 digits are divisible by }4.\)

\(\text{Option A:}\rightarrow\) \(\ 12\ ÷\ 4=3\ \ \checkmark\)
\(\text{Option B:}\rightarrow\) \(\ 32\ ÷\ 4=8\ \ \checkmark\)
\(\text{Option C:}\rightarrow\) \(\ 02\ \text{not divisible by 4}\)
\(\text{Option D:}\rightarrow\) \(\ 08\ ÷\ 4=2\ \ \checkmark\)

 
\(\Rightarrow C\)

Indices, SM-Bank 047 MC

Which of the following numbers is not divisible by 3?

  1. 3102
  2. 239
  3. 42
  4. 8121
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\(B\)

Show Worked Solution

\(\text{A number is divisible by }3\ \text{if the sum of its digits is divisible by }3.\)

\(\text{Option A:}\rightarrow\) \(3+1+0+2=6\ \ \checkmark\)
\(\text{Option B:}\rightarrow\) \(2+3+9=14\ \ \text{not divisible by 3}\)
\(\text{Option C:}\rightarrow\) \(4+2=6\ \ \checkmark\)
\(\text{Option D:}\rightarrow\) \(8+1+2+1=12\ \ \checkmark\)

 
\(\Rightarrow B\)

Indices, SM-Bank 046

Write \(324\) as a product of its prime factors in index form.  (2 marks)

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\(2^2\times3^4\)

Show Worked Solution
\(324\) \(=4\times 81\)
  \(=2\times 2\times 9\times 9\)
  \(=2\times 2\times 3\times 3\times 3\times 3\)
  \(=2^2\times3^4\)

Indices, SM-Bank 039

  1. Explain why a negative number raised to an odd power will always have a negative answer.  (2 marks)

  2. Give 2 worked examples that verify your explanation.  (2 marks)

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a.    \(\text{See worked solution}\)

b.    \(\text{See worked solution}\)

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a.    \(\text{A negative number multiplied by a negative}\)

\(\text{number an odd number of times always a has}\)

\(\text{negative answer.}\)

\(\text{It therefore, follows that when a negative number}\)

\(\text{is raised to an odd power, such as 1, 3, 5…, }\)

\(\text{the answer is also always negative.}\)

\(\text{ie } (-\times -)\times – =+\times -=-\)

b.    \(\text{Examples (Others are also acceptable)}\)

\((-1)^3=(-1\times -1)\times -1=1\times -1=-1\)

\((-1)^5=(-1\times -1)\times (-1\times -1)\times -1=1\times 1\times -1=-1\)

Indices, SM-Bank 038

  1. Explain why a negative number raised to an even power will always have a positive answer.  (2 marks)

  2. Give 2 worked examples that verify your explanation.  (2 marks)

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a.    \(\text{See worked solution}\)

b.    \(\text{See worked solution}\)

Show Worked Solution

a.    \(\text{A negative number multiplied by a negative}\)

\(\text{number always a has positive answer.}\)

\(\text{It therefore, follows that when a negative number}\)

\(\text{is multiplied by a negative number, an even}\)

\(\text{number of times, as is the case with even powers}\)

\(\text{such as 2, 4, 6…, the answer is also always positive.}\)

b.    \(\text{Examples (Others are also acceptable)}\)

\((-1)^2=-1\times -1=1\)

\((-1)^4=-1\times -1\times -1\times -1=1\)

Indices, SM-Bank 034

Evaluate \(2^2\times 7-5\times 4^3\).  (2 marks)

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\(-292\)

Show Worked Solution
\(2^2\times 7-5\times 4^3\) \(=(2\times 2)\times 7-5\times (4\times 4\times 4)\)
  \(=4\times 7-5\times 64\)
  \(=28-320=-292\)

Indices, SM-Bank 029

Write 625 in:

  1. Expanded form.  (1 mark)

  2. Index form.  (1 mark)

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a.    \(5\times 5\times 5\times 5\)

b.    \(5^4\)

Show Worked Solution
a.    \(625\) \(=5\times 125\)
    \(=5\times (5\times 25)\)
    \(=5\times 5\times 5\times 5\)

  

b.    \(625=5^4\)

 

Indices, SM-Bank 028

Write \(100\ 000\) in:

  1. Expanded form.  (1 mark)

  2. Index form.  (1 mark)

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a.    \(10 \times 10 \times 10 \times 10 \times 10\)

b.    \(10^5\)

Show Worked Solution

a.    \(100\ 000=10\times 10 \times 10 \times 10 \times 10\)

b.    \(100\ 000=10^5\)

 

Indices, SM-Bank 027

Write \(1\ 000\ 000\) in:

  1. Expanded form.  (1 mark)

  2. Index form.  (1 mark)

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a.    \(10 \times 10 \times 10 \times 10 \times 10 \times 10\)

b.    \(10^6\)

Show Worked Solution

a.    \(1\ 000\ 000=10\times 10 \times 10 \times 10 \times 10 \times 10\)

b.    \(1\ 000\ 000=10^6\)

Indices, SM-Bank 023

Insert the index powers that make this statement correct.  (2 marks)

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\(3\text{ and }2\)

Show Worked Solution
\(108\) \(=12\times 9\)
  \(=(4\times 3)\times (3\times 3)\)
  \(=(2\times 2\times 3)\times (3\times 3)\)
  \(=3^3\times 2^2\)

\(\therefore\ \text{The correct powers are }3\text{ and }2.\)

Indices, SM-Bank 022

Insert the index powers that make this statement correct.  (2 marks)

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\(2\text{ and }4\)

Show Worked Solution
\(144\) \(=12\times 12\)
  \(=(4\times 3)\times (4\times 3)\)
  \(=(2\times 2\times 3)\times (2\times 2\times 3)\)
  \(=3^2\times 2^4\)

\(\therefore\ \text{The correct powers are }2\text{ and }4.\)

Indices, SM-Bank 021

Insert the index power that makes this statement correct.  (2 marks)

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\(4\)

Show Worked Solution
\(81\) \(=9\times 9\)
  \(=(3\times 3)\times (3\times 3)\)
  \(=3^4\)

 
\(\therefore\ \text{The correct power is}\ 4\)

Indices, SM-Bank 020

Write  \(7\times 3\times 5\times 5\times 7\times 7\times 5\times 5\times 5\) in index form.  (2 marks)

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\(7^3\times 5^5\times 3^1\)

Show Worked Solution
\(7\times 3\times 5\times 5\times 7\times 7\times 5\times 5\times 5\) \(=7^3\times 5^5\times 3^1\)

Indices, SM-Bank 017

Write  \(27^2\times 2^3\) in expanded form.  (2 marks)

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\(3\times 3\times 3\times 3\times 3\times 3\times 2\times 2\times 2\)

Show Worked Solution
\(27^2\times 2^3\) \(=(27\times 27)\times(2\times 2\times 2)\)
  \(=(3\times 3\times 3)\times (3\times 3\times 3)\times (2\times 2\times 2)\)
  \(=3\times 3\times 3\times 3\times 3\times 3\times 2\times 2\times 2\)

Indices, SM-Bank 016

Write  \(3^3\times 49^2\) in expanded form.  (2 marks)

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\(3\times 3\times 3\times 7\times 7\times 7\times 7\)

Show Worked Solution
\(3^3\times 49^2\) \(= (3\times 3\times 3)\times (7\times 7)^2\)
  \(= (3\times 3\times 3) \times (7\times 7)\times (7\times 7)\)
  \(=3\times 3\times 3\times 7\times 7\times 7\times 7\)

Indices, SM-Bank 015 MC

Which expression is equal to  \(6^3\times 36^2\)?

  1. \(6\times 3\times 36\times 2\)
  2. \(6\times 6\times 6\times 6\times 6\)
  3. \(6\times 6\times 6\times 36\times 6\)
  4. \(6\times 6\times 6\times 6\times 6\times 6\times 6\)
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\(D\)

Show Worked Solution
\(6^3\times 36^2\) \(= (6\times 6\times 6)\times (6\times 6)^2\)
  \(= (6\times 6\times 6) \times (6\times 6)\times (6\times 6)\)
  \(=6\times 6\times 6\times 6\times 6\times 6\times 6\)

 
\(\Rightarrow D\)

Indices, SM-Bank 014 MC

\(2^5-2^3 =\)
 

Which number makes the expression above correct?

  1. \(4\)
  2. \(8\)
  3. \(24\)
  4. \(28\)
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\(C\)

Show Worked Solution
\(2^5-2^3\) \(=2\times 2\times 2\times 2\times 2-2\times 2\times 2\)
  \(=32-8\)
  \(=24\)

 
\(\Rightarrow C\)

Indices, SM-Bank 012 MC

\(30^2\) is equal to which of the following?

  1. \(60^2÷3\)
  2. \(3^2\times 2\times 5\times 2\times 5\)
  3. \(9\times 5^2\)
  4. \(3\times 10\times 10\)
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\(B\)

Show Worked Solution

\(30^2=900\)

\(\text{Checking Option B:}\)

\(3^2\times 2\times 5\times 2\times 5\)

\(= 3^2\times 10\times 10\)

\(=900\)

 
\(\Rightarrow B\)

Indices, SM-Bank 010 MC

Which one of these has the same value as \(16^2\)? 

  1. \(32^2\ ÷\ 4\)
  2. \(2\times 2\times 4\times 2\times 4\)
  3. \(2\times 8^2\)
  4. \(2\times 16\times 16\)
Show Answers Only

\(A\)

Show Worked Solution
\(16^2\) \(=16\times 16\)
  \(=32\ ÷\ 2\times 32\ ÷\ 2\)
  \(=32^2\ ÷\ 4\)

 
\(\Rightarrow A\)

Indices, SM-Bank 008

Write  \(20^2\times 32^1\times 16^4\)  in expanded form.  (2 marks)

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\(20\times 20\times 32\times 16\times 16\times 16\times 16\)

Show Worked Solution
\(20^2\times 32^1\times 16^4\) \(=20\times 20\times 32\times 16\times 16\times 16\times 16\)

Indices, SM-Bank 007

Write  \(7^1\times 5^4\times 4^3\)  in expanded form.  (2 marks)

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\(7\times 5\times 5\times 5\times 5\times 4\times 4\times 4\)

Show Worked Solution
\(7^1\times 5^4\times 4^3\) \(=7\times 5\times 5\times 5\times 5\times 4\times 4\times 4\)

Indices, SM-Bank 006

Write  \(3^3\times 2^3\)  in expanded form and evaluate.  (2 marks)

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\(3\times 3\times 3\times 2\times 2\times 2=216\)

Show Worked Solution
\(3^3\times 2^3\) \(=3\times 3\times 3\times 2\times 2\times 2\)
  \(=27\times 8=216\)