Squares and Square Roots — Exercise 3.1

Is the following natural number a perfect square? Give reason in support of your answer.

729

Answer

Finding prime factors of 729

$\begin{array}{l|r} 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

729 = (3 × 3) × (3 × 3) × (3 × 3)

Since 729 can be expressed as the product of pairs of equal prime factors.

Hence, 729 is a perfect square.

Is the following natural number a perfect square? Give reason in support of your answer.

5488

Answer

Finding prime factors of 5488

$\begin{array}{l|r} 2 & 5488 \\ \hline 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

5488 = (2 × 2) × (2 × 2) × (7 × 7) × 7

Since one 7 is left unpaired, so 5488 cannot be expressed as the product of pairs of equal prime factors.

Hence, 5488 is not a perfect square.

Is the following natural number a perfect square? Give reason in support of your answer.

1024

Answer

Finding prime factors of 1024

$\begin{array}{l|r} 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array}$

1024 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)

Since 1024 can be expressed as the product of pairs of equal prime factors.

Hence, 1024 is a perfect square.

Is the following natural number a perfect square? Give reason in support of your answer.

243

Answer

Finding prime factors of 243

$\begin{array}{l|r} 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

243 = (3 × 3) × (3 × 3) × 3

Since one 3 is left unpaired, so 243 cannot be expressed as the product of pairs of equal prime factors.

Hence, 243 is not a perfect square.

Show that the following number is a perfect square. Also find the number whose square is the given number.

1296

Answer

Finding prime factors of 1296

$\begin{array}{l|r} 2 & 1296 \\ \hline 2 & 648 \\ \hline 2 & 324 \\ \hline 2 & 162 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

1296 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3)

Since 1296 can be expressed as the product of pairs of equal prime factors, so 1296 is a perfect square.

1296 = (2)² × (2)² × (3)² × (3)²

⇒ 1296 = (2 × 2 × 3 × 3)²

⇒ 1296 = (36)²

Hence, 1296 is a perfect square and 36 is the number whose square is 1296.

Show that the following number is a perfect square. Also find the number whose square is the given number.

1764

Answer

Finding prime factors of 1764

$\begin{array}{l|r} 2 & 1764 \\ \hline 2 & 882 \\ \hline 3 & 441 \\ \hline 3 & 147 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

1764 = (2 × 2) × (3 × 3) × (7 × 7)

Since 1764 can be expressed as the product of pairs of equal prime factors, so 1764 is a perfect square.

1764 = (2)² × (3)² × (7)²

⇒ 1764 = (2 × 3 × 7)²

⇒ 1764 = (42)²

Hence, 1764 is a perfect square and 42 is the number whose square is 1764.

Show that the following number is a perfect square. Also find the number whose square is the given number.

3025

Answer

Finding prime factors of 3025

$\begin{array}{l|r} 5 & 3025 \\ \hline 5 & 605 \\ \hline 11 & 121 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array}$

3025 = (5 × 5) × (11 × 11)

Since 3025 can be expressed as the product of pairs of equal prime factors, so 3025 is a perfect square.

3025 = (5)² × (11)²

⇒ 3025 = (5 × 11)²

⇒ 3025 = (55)²

Hence, 3025 is a perfect square and 55 is the number whose square is 3025.

Show that the following number is a perfect square. Also find the number whose square is the given number.

3969

Answer

Finding prime factors of 3969

$\begin{array}{l|r} 3 & 3969 \\ \hline 3 & 1323 \\ \hline 3 & 441 \\ \hline 3 & 147 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

3969 = (3 × 3) × (3 × 3) × (7 × 7)

Since 3969 can be expressed as the product of pairs of equal prime factors, so 3969 is a perfect square.

3969 = (3)² × (3)² × (7)²

⇒ 3969 = (3 × 3 × 7)²

⇒ 3969 = (63)²

Hence, 3969 is a perfect square and 63 is the number whose square is 3969.

Find the smallest natural number by which 1008 should be multiplied to make it a perfect square.

Answer

Finding prime factors of 1008

$\begin{array}{l|r} 2 & 1008 \\ \hline 2 & 504 \\ \hline 2 & 252 \\ \hline 2 & 126 \\ \hline 3 & 63 \\ \hline 3 & 21 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

1008 = (2 × 2) × (2 × 2) × (3 × 3) × 7

Since the prime factor 7 is left unpaired, so to make 1008 a perfect square 7 should be paired.

So, the given number should be multiplied by 7.

Hence, the smallest natural number by which 1008 should be multiplied to make it a perfect square is 7.

Find the smallest natural number by which 5808 should be divided to make it a perfect square. Also, find the number whose square is the resulting number.

Answer

Finding prime factors of 5808

$\begin{array}{l|r} 2 & 5808 \\ \hline 2 & 2904 \\ \hline 2 & 1452 \\ \hline 2 & 726 \\ \hline 3 & 363 \\ \hline 11 & 121 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array}$

5808 = (2 × 2) × (2 × 2) × (11 × 11) × 3

Since the prime factor 3 is left unpaired, so to make 5808 a perfect square it should be divided by 3.

$\Rightarrow 5808 \div 3 = 1936\\[1em] \Rightarrow 1936 = (2 \times 2) \times (2 \times 2) \times (11 \times 11)\\[1em] \Rightarrow \sqrt{1936} = 2 \times 2 \times 11\\[1em] \Rightarrow \sqrt{1936} = 44$

Hence, the smallest natural number by which 5808 should be divided to make it a perfect square is 3 and 44 is the number whose square is the resulting number 1936.