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<p>The Singapore Flyer is a giant observation wheel with seating capacity of $784$ passengers.</p>
<p>The number of passengers that each capsule can carry is equal to the total number of capsules on the Singapore Flyer.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the prime factorisation of $784$ and express your answer in index notation.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the total number of capsules on the Singapore Flyer.</span></div></li></ol>

The Singapore Flyer is a giant observation wheel with seating capacity of $784$ passengers.

The number of passengers that each capsule can carry is equal to the total number of capsules on the Singapore Flyer.

Find the prime factorisation of $784$ and express your answer in index notation.

Hence, find the total number of capsules on the Singapore Flyer.

Square and Cube Roots Using Prime Factorisation

Prime Factorisation of Positive Integers

IP1U1 SWIII Q5

Primes, HCF and LCM

<p>The number $1728$ written as a product of its prime factors is $1728 = 2^{6} \times 3^{3}$.</p>
<p>Using only the above result and showing your working clearly, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the smallest positive integer value of $k$ for which $1728k$ is a square number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">an expression in terms of $x$ for $\sqrt[3]{1728x^9}$.</span></div></li></ol>

The number $1728$ written as a product of its prime factors is $1728 = 2^{6} \times 3^{3}$.

Using only the above result and showing your working clearly, find

the smallest positive integer value of $k$ for which $1728k$ is a square number,

an expression in terms of $x$ for $\sqrt[3]{1728x^9}$.

Perfect Squares and Cubes Using Prime Factors

IP1U1 SWII Q14

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $2744$ as the product of its prime factors. Give your answer in index notation.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the smallest positive integer value of $n$, for&nbsp;$\frac{2744}{n}$ to be a perfect square.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your answer from part <strong>(a)</strong> to find the value of $\sqrt[3]{2744}$.</div></li></ol>

Express $2744$ as the product of its prime factors. Give your answer in index notation.

IP1U1 SWII Q13

<p>Expressed as the product of its prime factors, $3375=3^{3} \times 5^{3}$ and $32400=2^{4} \times 3^{4} \times 5^{2}$.</p>
<p>Using these results,</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">find in prime factorization form, the highest common factor of $3375$ and $32400$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">find the value of $\sqrt{32400} \div \sqrt[3]{3375}$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">find the smallest positive integer $k$, such that&nbsp;$3375k$ is a perfect square.</div></li></ol>

Expressed as the product of its prime factors, $3375=3^{3} \times 5^{3}$ and $32400=2^{4} \times 3^{4} \times 5^{2}$.

find in prime factorization form, the highest common factor of $3375$ and $32400$.

find the value of $\sqrt{32400} \div \sqrt[3]{3375}$.

find the smallest positive integer $k$, such that $3375k$ is a perfect square.

Highest Common Factor and Its Applications

IP1U1 SWII Q10

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $2744$ as a product of its prime factors, giving your answer in index notation.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence, find $\sqrt[3]{2744}$ without the use of a calculator.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.</div></li></ol>

Express $2744$ as a product of its prime factors, giving your answer in index notation.

Hence, find $\sqrt[3]{2744}$ without the use of a calculator.

Given that $2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.

IP1U1 SWII Q9

<p>Without using a calculator and showing all your working clearly, </p>
<p>evaluate $\sqrt{2^{18} \times 5^6 \times 11^4}$, giving your answer in index notation.</p>

Without using a calculator and showing all your working clearly, 

evaluate $\sqrt{2^{18} \times 5^6 \times 11^4}$, giving your answer in index notation.

IP1U1 SWI Q13

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $343000 = 2^x \times 5^y \times 7^z$, find the values of $x$, $y$ and $z$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, determine the cube root of $343000$.</span></div></li></ol>

Given that $343000 = 2^x \times 5^y \times 7^z$, find the values of $x$, $y$ and $z$.

Hence, determine the cube root of $343000$.

IP1U1 SWI Q12

<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\sqrt[3]{125\ 000}$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sqrt[3]{250\ 047}$,</span></div></li></ol>
<p>by using prime factorisation.</p>

Prime Factorisation and Its Applications

<p>Using prime factorisation, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\sqrt{90\ 000}$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sqrt{99\ 225}$,</span></div></li></ol>

 Prime Factorisation and Its Applications

<ol class="abc-list"><li data-list-item-id="item-1763490717016-s92sd7g3x" data-list-item-path="item-1763490717016-s92sd7g3x"><div class="list-item-paragraph">Express $2744$ as a product of its prime factors, giving your answer in index notation.</div></li><li data-list-item-id="item-1763490717729-k6fh50a5w" data-list-item-path="item-1763490717729-k6fh50a5w"><div class="list-item-paragraph">Hence, find $\sqrt[3]{2744}$ without the use of a calculator.</div></li><li data-list-item-id="item-1763490718455-c5ctwrsvj" data-list-item-path="item-1763490718455-c5ctwrsvj"><div class="list-item-paragraph">Given that&nbsp;$2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.</div></li></ol>

Prime Factorisation, Cube Roots, and Perfect Square Conditions

<ol class="abc-list"><li><div class="list-item-paragraph">Express $74088$ as a product of prime factors.</div></li><li><div class="list-item-paragraph">Hence, or otherwise, find the cube root of $74088$.</div></li></ol>

Prime Factorisation and Cube Root Calculation

<ol class="abc-list"><li><div class="list-item-paragraph">Express $5832$ as a product of its prime factors, giving your answer in index notation.</div></li><li><div class="list-item-paragraph">Hence, without using the calculator, evaluate $\sqrt[3]{5832}$.</div></li></ol>

Prime Factorisation and Cube Root Evaluation

<ol class="abc-list"><li><div class="list-item-paragraph">Express $784$ as a product of its prime factors, giving your answer in index notation.</div></li><li><div class="list-item-paragraph">Hence, find the value of $\sqrt{78400}$.</div></li></ol>

Prime Factorisation and Square Root Evaluation

<p>The dimensions of a rectangle are $({{2}^{2}}\times {{5}^{2}}\times 7)$cm by $({{5}^{2}}\times {{7}^{3}})$cm.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the area of the rectangle. Express your answer in prime factorization form.</div></li><li><div class="list-item-paragraph">Given that a square has the same area as the rectangle in part <strong>(a)</strong>. Find the length of the square side.</div></li></ol>

The dimensions of a rectangle are $({{2}^{2}}\times {{5}^{2}}\times 7)$cm by $({{5}^{2}}\times {{7}^{3}})$cm.

Perimeter and Area of Rectangles and Squares

Prime Factorisation in Area and Geometry Problems

Perimeter and Area of Plane Figures

<p>Expressed as the product of its prime factors, $240={{2}^{4}}\times 3\times 5$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $7056$ as a product of prime factors.</div></li><li><div class="list-item-paragraph">Use your answer in <strong>(a)</strong> to find the positive square root of $7056$, leaving your answer in index notation.</div></li><li><div class="list-item-paragraph">Find the smallest positive integer $n$ for which $\sqrt{240n}$ is a whole number.</div></li></ol>

Expressed as the product of its prime factors, $240={{2}^{4}}\times 3\times 5$.

Prime Factorisation and Square Roots

<ol class="i-list"><li data-list-item-id="item-1760761288899-01inzbbnm" data-list-item-path="item-1760761288899-01inzbbnm"><div class="list-item-paragraph">Find $\sqrt{627264}$ by using prime factorisation.</div></li><li data-list-item-id="item-1760761289297-dzuriwx5m" data-list-item-path="item-1760761289297-dzuriwx5m"><div class="list-item-paragraph">Find $\sqrt[3]{250047}$ using prime factorisation.</div></li><li data-list-item-id="item-1760761290242-rw09prgug" data-list-item-path="item-1760761290242-rw09prgug"><div class="list-item-paragraph">Find $\sqrt{15376}$ using prime factorisation.</div></li></ol>

Finding Square and Cube Roots Using Prime Factorisation

<ol class="abc-list"><li data-list-item-id="item-1758604980745-s612khero" data-list-item-path="item-1758604980745-s612khero"><div class="list-item-paragraph">Express $2744$ as a product of its prime factors, giving your answer in index notation.</div></li><li data-list-item-id="item-1758604981271-ojcnaboxe" data-list-item-path="item-1758604981271-ojcnaboxe"><div class="list-item-paragraph">Hence, find $\sqrt[3]{2744}$ without the use of a calculator.</div></li><li data-list-item-id="item-1758604982238-9kts9qrtr" data-list-item-path="item-1758604982238-9kts9qrtr"><div class="list-item-paragraph">Given that&nbsp;$2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.</div></li></ol>

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