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

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

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)

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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)

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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.    \(2^9\)

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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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\(2^7\)

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\(a^m\ ÷\ a^n=a^{m-n}\)

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

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

Indices, SM-Bank 091

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

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\(3^{11}\)

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\(a^m\ ÷\ a^n=a^{m-n}\)

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

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

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)

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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)

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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.    \(3^2\)

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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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\(3^7\)

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

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

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

Indices, SM-Bank 085

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

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\(2^6\)

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

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

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

Indices, SM-Bank 084

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

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\(4^5\)

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

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

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

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.   \(3^2\times 3^4=3^6\)