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Indices, SM-Bank 101

Simplify  \(2^3\times 5^4\times 2^5\ ÷\  5^2\), giving your answer in simplified index form.  (2 marks)

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\(2^8\times 5^2\)

Show Worked Solution
\(2^3\times 5^4\times 2^5\ ÷\  5^2\) \(=2^{3+5}\times 5^{4-2}\)
  \(=2^8\times 5^2\)

Indices, SM-Bank 099

Simplify the following, giving your answers in index form.

  1. \((2+3)^0\)  (1 mark)

  2. \(2(4^0)-2\)  (2 marks)

  3. \(3\times 6^0+4\times 2^0\)  (2 marks)

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a.    \(1\)

b.    \(0\)

b.    \(7\)

Show Worked Solution

a.   \((2+3)^0=5^0=1\)

b.   \(2(4^0)-2=2\times 1-2=0\)

c.   \(3\times 6^0+4\times 2^0=3\times 1+4\times 1=7\)

Indices, SM-Bank 098

Simplify the following, giving your answers in index form.

  1. \(4^0\)  (1 mark)

  2. \(6^0+2^0\)  (2 marks)

  3. \(4+2^0\)  (2 marks)

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a.    \(1\)

b.    \(2\)

b.    \(5\)

Show Worked Solution

a.   \(4^0=1\)

b.   \(6^0+2^0=1+1=2\)

c.   \(4+2^0=4+1=5\)

Indices, SM-Bank 097

Simplify the following, giving your answers in index form.

  1. \(4^2\ ÷\ 4^2\)  (1 mark)

  2. \(6^3\ ÷\ 6^3\)  (1 mark)

  3. \(12^7\ ÷\ 12^7\)  (1 mark)

Show Answers Only

a.    \(4^0\)

b.    \(6^0\)

b.    \(12^0\)

Show Worked Solution

a.   \(4^2\ ÷\ 4^2=4^{2-2}=4^0\)

b.   \(6^3\ ÷\ 6^3=6^{3-3}=6^0\)

c.   \(12^7\ ÷\ 12^7=12^{7-7}=12^0\)

Indices, SM-Bank 096

Simplify the following, giving your answers in index form.

  1. \((5^2)^2\)  (1 mark)

  2. \((6^3)^4\)  (1 mark)

  3. \((7^4)^5\)  (1 mark)

Show Answers Only

a.    \(5^4\)

b.    \(6^{12}\)

b.    \(7^{20}\)

Show Worked Solution

a.   \((5^2)^2=5^{2\times 2}=5^4\)

b.   \((6^3)^4=6^{3\times 4}=6^{12}\)

c.   \((7^4)^5=7^{4\times 5}=7^{20}\)

Indices, SM-Bank 095

Simplify the following, giving your answers in index form.

  1. \((2^4)^4\)  (1 mark)

  2. \((4^3)^5\)  (1 mark)

  3. \((3^3)^3\)  (1 mark)

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a.    \(2^{16}\)

b.    \(4^{15}\)

b.    \(3^9\)

Show Worked Solution

a.   \((2^4)^4=2^{4\times 4}=2^{16}\)

b.   \((4^3)^5=4^{3\times 5}=4^{15}\)

c.   \((3^3)^3=3^{3\times 3}=3^9\)

Indices, SM-Bank 094

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

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

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a.    \((5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)\)

b.    \(5^8\)

Show Worked Solution

a.   \((5^2)^4=(5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)\)
 

b.   \((5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)=5^8\)

Indices, SM-Bank 093

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

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

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

b.    \(2^9\)

Show Worked Solution

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

b.   \((2\times 2\times 2)\times (2\times 2\times 2)\times (2\times 2\times 2)=2^9\)

Indices, SM-Bank 092

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

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\(2^7\)

Show Worked Solution

\(a^m\ ÷\ a^n=a^{m-n}\)

\(2^n\ ÷\ 2^2\) \(=2^5\)
\(\therefore \ n-2\) \(=5\)
\(n\) \(=7\)

 
\(\therefore \ \text{the missing term is }2^7\)

Indices, SM-Bank 091

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

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\(3^{11}\)

Show Worked Solution

\(a^m\ ÷\ a^n=a^{m-n}\)

\(3^n\ ÷\ 3^4\) \(=3^7\)
\(\therefore \ n-4\) \(=7\)
\(n\) \(=11\)

 
\(\therefore \ \text{the missing term is }3^{11}\)

Indices, SM-Bank 090

Simplify the following, giving your answers in index form.

  1. \(\dfrac{2^{16}}{2^9}\)  (1 mark)

  2. \(\dfrac{4^8}{4^3}\)  (1 mark)

  3. \(\dfrac{10^{11}}{10^2}\)  (1 mark)

Show Answers Only

a.    \(2^7\)

b.    \(4^5\)

b.    \(10^9\)

Show Worked Solution

a.   \(\dfrac{2^{16}}{2^9}=2^{16-9}=2^7\)

b.   \(\dfrac{4^8}{4^3}=4^{8-3}=4^5\)

c.   \(\dfrac{10^{11}}{10^2}=10^{11-2}=10^9\)

Indices, SM-Bank 089

Simplify the following, giving your answers in index form.

  1. \(2^8 \ ÷\ 2^4\)  (1 mark)

  2. \(5^{15} \ ÷\ 5^9\)  (1 mark)

  3. \(3^4 \ ÷\ 3\)  (1 mark)

Show Answers Only

a.    \(2^4\)

b.    \(5^6\)

b.    \(3^3\)

Show Worked Solution

a.   \(2^8\ ÷\ 2^4=2^{8-4}=2^4\)

b.   \(5^{15}\ ÷\ 5^9=5^{15-9}=5^6\)

c.   \(3^4\ ÷\ 3^1=3^{4-1}=3^3\)

Indices, SM-Bank 088

  1. Write  \(\dfrac{7^5}{7^2}\)  as a fraction in expanded form.  (2 marks)

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

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a.    \(\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}\)

b.    \(7^3\)

Show Worked Solution

a.   \(\dfrac{7^5}{7^2}=\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}\)
 

b.   \(\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}=7^3\)

Indices, SM-Bank 087

  1. Write  \(\dfrac{3^6}{3^4}\)  as a fraction in expanded form.  (2 marks)

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

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a.    \(\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}\)

b.    \(3^2\)

Show Worked Solution

a.   \(\dfrac{3^6}{3^4}=\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}\)
 

b.   \(\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}=3^2\)

Indices, SM-Bank 086

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

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\(3^7\)

Show Worked Solution

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

\(3^n\times 3^4\) \(=3^{11}\)
\(\therefore \ n+4\) \(=11\)
\(n\) \(=7\)

 
\(\therefore \ \text{the missing term is }3^7\)

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

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. \(8\ \text{and }9\)
Show Answers Only

\(D\)

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. \(6\ \text{and }7\)
Show Answers Only

\(C\)

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)

Show Answers Only

\(\text{See worked solution}\)

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)

Show Answers Only

\(\text{See worked solution}\)

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. \(6\)
Show Answers Only

\(C\)

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. \(578\)
Show Answers Only

\(C\)

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. \(194\ 481\)
Show Answers Only

\(A\)

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. \(21.3\)
Show Answers Only

\(B\)

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. \(171\)
Show Answers Only

\(A\)

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. \(5\)
Show Answers Only

\(D\)

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

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)

Show Answers Only

\(\text{See worked solution}\)

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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\(\text{See worked solution}\)

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

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

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

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

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

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

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

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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\(\text{See worked solution}\)

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

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

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

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

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

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

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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\(\text{See worked solution}\)

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

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

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

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

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

Indices, SM-Bank 052

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

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\(\text{See worked solution}\)

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

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

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

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

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

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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\(C\)

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

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

 
\(\Rightarrow C\)

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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\(C\)

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

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

 
\(\Rightarrow C\)

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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\(A\)

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

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

 
\(\Rightarrow A\)

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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\(C\)

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

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

 
\(\Rightarrow C\)

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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\(B\)

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

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

 
\(\Rightarrow B\)

Indices, SM-Bank 046

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

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\(2^2\times3^4\)

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