2 Marks Questions

Question 12 Marks

64 is a square number $\left(8^2\right)$ and a cube number $\left(4^3\right)$. Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Answer

Yes, there are other numbers that are both squares and cubes. For example :
$\begin{array}{l}729=9^3 \text { (cube number) }=27^2 \text { (perfect square) } \\4096=16^3 \text { (cube number) }=64^2 \text { (perfect square) }\end{array}$
General Rule: The sixth power of any number (i.e., $n^6$ ) is both a square and a cube. i.e.
$\begin{array}{l}1^6=1 \\2^6=64 \\3^6=729 \\4^6=4096 \\5^6=15,625\end{array}$

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Question 22 Marks

Circle the numbers that are the same -
$2^4 \times 3^6, 6^4 \times 3^2, 6^{10}, 18^2 \times 6^2, 6^{24}$

Answer

(i) $2^4 \times 3^6$
(ii) $6^4 \times 3^2=(2 \times 3)^4 \times 3^2=2^4 \times 3^4 \times 3^2=2^4 \times 3^6$
(iii) $6^{10}=(2 \times 3)^{10}=2^{10} \times 3^{10}$
(iv) $18^2 \times 6^2=(2 \times 3 \times 3)^2 \times(2 \times 3)^2=2^2 \times 3^2 \times 3^2 \times 2^2 \times 3^2=2^4 \times 3^6$
(v) $6^{24}=(2 \times 3)^{24}=2^{24} \times 3^{24}$
Therefore, $2^4 \times 3^6, 64 \times 32$, and $18^2 \times 6^2$ are the same.

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Question 32 Marks

Write the given number as the product of two or more powers in three different ways. The powers can be any integers. $32^{-5}$

Answer

$32^{-5}$
$\begin{array}{l}32=2 \times 2 \times 2 \times 2 \times 2=2^5 \\32^{-5}=\left(2^5\right)^{-5}=2^{-25}\end{array}$
Three different ways :
1. $32^{-5}=2^{-25}=2^{-15} \times 2^{-10}$
2. $32^{-5}=2^{-25}=2^{-5} \times 2^{-5} \times 2^{-5} \times 2^{-5} \times 2^{-5}$
3. $32^{-5}=2^{-25}=2^{-5} \times 2^{-20}$

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Question 42 Marks

Write the given number as the product of two or more powers in three different ways. The powers can be any integers. $192^8$

Answer

$192^8$
$\begin{array}{l}192=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3=2^6 \times 3 \\192^8=\left(2^6 \times 3\right)^8=2^{6 \times 8} \times 3^8=2^{48} \times 3^8\end{array}$
Three different ways :
1. $192^8=2^{48} \times 3^8=2^{40} \times 2^8 \times 3^8=2^{40} \times(2 \times 3)^8=2^{40} \times 6^8$
2. $192^8=2^{48} \times 3^8=\left(2^{24} \times 3^4\right) \times\left(2^{24} \times 3^4\right)$
3. $192^8=2^{48} \times 3^8=\left(2^6\right)^8 \times 3^8$

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Question 52 Marks

Write the given number as the product of two or more powers in three different ways. The powers can be any integers. $64^3$

Answer

$64^3$
$\begin{array}{l}64=2 \times 2 \times 2 \times 2 \times 2 \times 2=2^6 \\64^3=\left(2^6\right)^3=2^{18}\end{array}$
Three different ways :
1. $64^3=2^{18}=2^9 \times 2^9$
2. $64^3=2^{18}=2^{10} \times 2^8$
3. $64^3=2^{18}=2^6 \times 2^6 \times 2^6$

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Question 62 Marks

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would be there after 40 days?

Answer

Number of containers added every day = 1
Number of containers after 40 days = 40 containers
Number of bottles in a container = 5
Total bottles in 40 containers = 5 × 40 = 200 bottles
Therefore, there would be 200 bottles after 40 days.

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