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Indices, SM-Bank 085

Find the missing term in the following number sentence.  (1 mark)

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\(2^6\)

Show Worked Solution

\(a^m\times a^n=a^{m+n}\)

\(2^8\times 2^n\) \(=2^{14}\)
\(\therefore \ 8+n\) \(=14\)
\(n\) \(=6\)

 
\(\therefore \ \text{the missing term is }2^6\)

Indices, SM-Bank 084

Find the missing term in the following number sentence.  (1 mark)

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\(4^5\)

Show Worked Solution

\(a^m\times a^n=a^{m+n}\)

\(4^3\times 4^n\) \(=4^8\)
\(\therefore \ 3+n\) \(=8\)
\(n\) \(=5\)

 
\(\therefore \ \text{the missing term is }4^5\)

Indices, SM-Bank 083

Simplify the following, giving your answers in index form.

  1. \(2^5\times 2^4\times 2\)  (1 mark)

  2. \(4^3\times 4^2\times 4^6\)  (1 mark)

  3. \(3^2\times 3\times 3^3\)  (1 mark)

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a.    \(2^{10}\)

b.    \(4^{11}\)

b.    \(3^6\)

Show Worked Solution

a.   \(2^5\times 2^4\times 2=2^{5+4+1}=2^{10}\)

b.   \(4^3\times 4^2\times 4^6=4^{3+2+6}=4^{11}\)

c.   \(3^2\times 3^1\times 3^3=3^{2+1+3}=3^6\)

Indices, SM-Bank 082

Simplify the following, giving your answers in index form.

  1. \(5^3\times 5^2\)  (1 mark)

  2. \(7^7\times 7\)  (1 mark)

  3. \(6^4\times 6^3\)  (1 mark)

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a.    \(5^5\)

b.    \(7^8\)

b.    \(6^7\)

Show Worked Solution

a.   \(5^3\times 5^2=5^{3+2}=5^5\)

b.   \(7^7\times 7^1=7^{7+1}=7^8\)

c.   \(6^4\times 6^3=6^{4+3}=6^7\)

Indices, SM-Bank 081

  1. Write \(2^3\times 2^2\) in expanded form.  (1 mark)

  2. Write your answer to (a) in index form.  (1 mark)

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

b.    \(2^5\)

Show Worked Solution

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

b.   \(2^3\times 2^2=2^5\)

Indices, SM-Banks 080

  1. Write \(3^2\times 3^4\) in expanded form.  (1 mark)

  2. Write your answer to (a) in index form.  (1 mark)

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

b.    \(3^6\)

Show Worked Solution

a.   \(3^2\times 3^4=(3\times 3)\times (3\times 3\times 3\times 3)\)

b.   \(3^2\times 3^4=3^6\)

Indices, SM-Bank 079

Evaluate  \(3\times\sqrt{81}+\sqrt[3]{27}\times 2\).  (2 marks)

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\(33\)

Show Worked Solution
\(3\times \sqrt{81}+\sqrt[3]{27}\times 2\) \(=3\times\sqrt{9\times 9}+\sqrt[3]{3\times 3\times 3}\times 2\)
  \(=3\times 9+3\times 2\)
  \(=33\)

Indices, SM-Bank 078

Evaluate \(3\times\sqrt[3]{64}-\sqrt{100}\).  (2 marks)

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\(2\)

Show Worked Solution
\(3\times \sqrt[3]{64}-\sqrt{100}\) \(=3\times\sqrt[3]{4\times 4\times 4}-\sqrt{10\times 10}\)
  \(=3\times 4-10\)
  \(=2\)

Indices, SM-Bank 075

Find the missing whole number that makes the following number sentence correct.  (2 marks)

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Show Worked Solution

\(\text{Using trial and error method:}\)

\(\text{Test 1st square number}\ \rightarrow 1\)

\(\text{LHS:}\) \(\ 3\times 1^2+(\sqrt{25}-1)=3\times 1+(5-1)\)
  \(=7\)
\(\text{RHS:}\) \(\ \sqrt{1}\times 9\times 3-5=1\times 27-5\)
  \(=23\)

\(\text{LHS }\ne\text{ RHS}\)
 

\(\text{Test 2nd square number}\ \rightarrow 4\)

\(\text{LHS:}\) \(\ 3\times 4^2+(\sqrt{25}-4)=3\times 16+(5-4)\)
  \(=49\)
\(\text{RHS:}\) \(\ \sqrt{4}\times 9\times 3-5=2\times 27-5\)
  \(=49\)

\(\text{LHS }=\text{ RHS}\)

\(\therefore \ \)

 

Indices, SM-Bank 074 MC

Between which two number does \(\sqrt{70}\)  lie?

  1. \(5\ \text{and }6\)
  2. \(6\ \text{and }7\)
  3. \(7\ \text{and }8\)
  4. \(8\ \text{and }9\)
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\(D\)

Show Worked Solution
\(\text{Consider Option D:}\) \(\rightarrow\ \sqrt{64}=8\)
  \(\rightarrow\ \sqrt{81}=9\)

 

\(\therefore\ \sqrt{70}\ \text{lies between }\sqrt{64} \text{ and }\sqrt{81}\)

\(\therefore\ \sqrt{70}\ \text{lies between }8 \text{ and }9\)

\(\Rightarrow D\)

Indices, SM-Bank 073 MC

Between which two number does \(\sqrt{30}\) lie?

  1. \(3\ \text{and }4\)
  2. \(4\ \text{and }5\)
  3. \(5\ \text{and }6\)
  4. \(6\ \text{and }7\)
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\(C\)

Show Worked Solution
\(\text{Consider Option C:}\) \(\rightarrow\ \sqrt{25}=5\)
  \(\rightarrow\ \sqrt{36}=6\)

 

\(\therefore\ \sqrt{30}\ \text{lies between }\sqrt{25} \text{ and }\sqrt{36}\)

\(\therefore\ \sqrt{30}\ \text{lies between }5 \text{ and }6\)

 
\(\Rightarrow C\)

Indices, SM-Bank 072

Show that  \(\sqrt{9\times 4}=\sqrt{9}\times \sqrt{4}\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution
\(\text{LHS}:\sqrt{9\times 4}\) \(=\sqrt{ 3\times 3\times 2\times 2}\)
  \(=\sqrt{ 6\times 6}\)
  \(=6\)

 

\(\text{RHS}:\sqrt{9}\times \sqrt{4}\) \(=3\times 2\)
  \(=6\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{9\times 4}=\sqrt{9}\times \sqrt{4}\)

Indices, SM-Bank 071

Show that  \(\sqrt{25\times 16}=\sqrt{25}\times \sqrt{16}\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution
\(\text{LHS}:\sqrt{25\times 16}\) \(=\sqrt{ 5\times 5\times 4\times 4}\)
  \(=\sqrt{ 20\times 20}\)
  \(=20\)

 

\(\text{RHS}:\sqrt{25}\times \sqrt{16}\) \(=5\times 4\)
  \(=20\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{25\times 16}=\sqrt{25}\times \sqrt{16}\)

Indices, SM-Bank 070

Show that  \(\sqrt{225}=\sqrt{25}\times \sqrt{9}\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution
\(\text{LHS}:\sqrt{225}\) \(=\sqrt{ 5\times 45}\)
  \(=\sqrt{ 5\times 5\times 9}\)
  \(=\sqrt{ 5\times 5\times 3\times 3}\)
  \(=\sqrt{ 15\times 15}\)
  \(=15\)

 

\(\text{RHS}:\sqrt{25}\times \sqrt{9}\) \(=5\times 3\)
  \(=15\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{225}=\sqrt{25}\times \sqrt{9}\)

Indices, SM-Bank 069

Show that  \(\sqrt{144}=\sqrt{36}\times \sqrt{4}\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution
\(\text{LHS}:\sqrt{144}\) \(=\sqrt{ 12\times 12}\)
  \(=12\)

 

\(\text{RHS}:\sqrt{36}\times \sqrt{4}\) \(=6\times 2\)
  \(=12\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{144}=\sqrt{36}\times \sqrt{4}\)

Indices, SM-Bank 068 MC

Given that  \(12^2=144\), then \(\sqrt{144}=\) ?

  1. \(288\)
  2. \(72\)
  3. \(12\)
  4. \(6\)
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\(C\)

Show Worked Solution

\(144=12\times 12\ \ \text{(Given)}\)

\(\therefore \sqrt{144}\) \(=\sqrt{ 12\times 12}\)
  \(=12\)

 
\(\Rightarrow C\)

Indices, SM-Bank 067 MC

Given that  \(17^2=289\), then \(\sqrt{289}=\) ?

  1. \(8.5\)
  2. \(13.5\)
  3. \(17\)
  4. \(578\)
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\(C\)

Show Worked Solution

\(289=17\times 17\ \ \text{(Given)}\)

\(\therefore \sqrt{289}\) \(=\sqrt{ 17\times 17}\)
  \(=17\)

 
\(\Rightarrow C\)

Indices, SM-Bank 066 MC

Given that  \(21^2=441\), then \(\sqrt{441}=\) ?

  1. \(21\)
  2. \(42\)
  3. \(420\)
  4. \(194\ 481\)
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\(A\)

Show Worked Solution

\(441=21\times 21\ \ \text{(Given)}\)

\(\therefore \sqrt{441}\) \(=\sqrt{ 21\times 21}\)
  \(=21\)

 
\(\Rightarrow A\)

Indices, SM-Bank 065 MC

Given that  \(4^3=64\), then \(\sqrt[3]{64}=\) ?

  1. \(2\)
  2. \(4\)
  3. \(8\)
  4. \(21.3\)
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\(B\)

Show Worked Solution

\(64=4\times 4\times 4\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{64}\) \(=\sqrt[3]{ 4\times 4\times 4}\)
  \(=4\)

 
\(\Rightarrow B\)

Indices, SM-Bank 064 MC

Given that  \(8^3=512\), then \(\sqrt[3]{512}=\) ?

  1. \(8\)
  2. \(23\)
  3. \(128\)
  4. \(171\)
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\(A\)

Show Worked Solution

\(512=8\times 8\times 8\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{512}\) \(=\sqrt[3]{ 8\times 8\times 8}\)
  \(=8\)

 
\(\Rightarrow A\)

Indices, SM-Bank 063 MC

Given that  \(5^3=125\), then \(\sqrt[3]{125}=\) ?

  1. \(62.5\)
  2. \(41.7\)
  3. \(11.2\)
  4. \(5\)
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\(D\)

Show Worked Solution

\(125=5\times 5\times 5\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{125}\) \(=\sqrt[3]{ 5\times 5\times 5}\)
  \(=5\)

 
\(\Rightarrow D\)

Indices, SM-Bank 062

  1. Write 900 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt{900}\).  (2 marks)

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a.    \(2\times 2\times 3\times 3\times 5\times 5\)

b.    \(30\)

Show Worked Solution
a.    \(900\) \(=9\times 100\)
    \(=3\times 3\times 10\times 10\)
    \(=3\times 3\times 2\times 5\times 2\times 5\)
    \(=2\times 2\times 3\times 3\times 5\times 5\)

 

b.    \(\sqrt{900}\) \(=\sqrt{2\times 2\times 3\times 3\times 5\times 5}\)
    \(=\sqrt{(2\times 3\times 5)\times (2\times 3\times 5)}\)
    \(=\sqrt{30\times 30}\)
    \(=30\)

Indices, SM-Bank 061

  1. Write 1024 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt{1024}\).  (2 marks)

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

b.    \(32\)

Show Worked Solution
a.    \(1024\) \(=2\times 512\)
    \(=2\times 2\times 256\)
    \(=2\times 2\times 2\times 128\)
    \(=2\times 2\times 2\times 2\times 64\)
    \(=2\times 2\times 2\times 2\times 8\times 8\)
    \(=2\times 2\times 2\times 2\times 2\times 4\times 2\times 4\)
    \(=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

 

b.    \(\sqrt{1024}\) \(=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}\)
    \(=\sqrt{(2\times 2\times 2\times 2\times 2)\times (2\times 2\times 2\times 2\times 2)}\)
    \(=\sqrt{32\times 32}\)
    \(=32\)

Indices, SM-Bank 060

  1. Write 324 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt{324}\).  (2 marks)

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

b.    \(18\)

Show Worked Solution
a.    \(324\) \(=2\times 162\)
    \(=2\times 2\times 81\)
    \(=2\times 2\times 9\times 9\)
    \(=2\times 2\times 3\times 3\times 3\times 3\)

 

b.    \(\sqrt{324}\) \(=\sqrt{2\times 2\times 3\times 3\times 3\times 3}\)
    \(=\sqrt{(2\times 3\times 3)\times (2\times 3\times 3)}\)
    \(=\sqrt{18\times 18}\)
    \(=18\)

Indices, SM-Bank 059

  1. Write 256 as a product of its prime factors.  (2 marks)

  2. Hence find \(\sqrt{256}\).  (2 marks)

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

b.    \(16\)

Show Worked Solution
a.    \(256\) \(=2\times 128\)
    \(=2\times 2\times 64\)
    \(=2\times 2\times 2\times 32\)
    \(=2\times 2\times 2\times 2\times 16\)
    \(=2\times 2\times 2\times 2\times 2\times 8\)
    \(=2\times 2\times 2\times 2\times 2\times 2\times 4\)
    \(=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

 

b.    \(\sqrt{256}\) \(=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}\)
    \(=\sqrt{(2\times 2\times 2\times 2)\times (2\times 2\times 2\times 2)}\)
    \(=\sqrt{16\times 16}\)
    \(=16\)

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
Show Answers Only

\(C\)

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. \(1\ 202\ 005\)
Show Answers Only

\(C\)

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
Show Answers Only

\(A\)

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
Show Answers Only

\(C\)

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
Show Answers Only

\(B\)

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.    \(\text{See worked solution}\)

Show Worked Solution

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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\(-292\)

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

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.    \(5^4\)

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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.    \(10^5\)

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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.    \(10^6\)

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

b.    \(1\ 000\ 000=10^6\)