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Find the square root of the following: (c) 169000 is a perfect square number. 4. Find the square root of the following method: (a) 169 (b) 324 (c) 1225

Mathematics

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Answers

3 parts
a

13\text{13}

b

18\text{18}

c

35\text{35}

Step-by-step solution

6 steps

Here are the solutions for the problems:

3. (c) 169000 is perfect square number.

To determine if a number is a perfect square, we check its prime factorization. If all the exponents in the prime factorization are even, then the number is a perfect square.

Step 1: Find the prime factorization of 169000.

169000=169×1000169000 = 169 \times 1000 169=13×13=132169 = 13 \times 13 = 13^2 1000=10×10×10=(2×5)×(2×5)×(2×5)=23×531000 = 10 \times 10 \times 10 = (2 \times 5) \times (2 \times 5) \times (2 \times 5) = 2^3 \times 5^3

So,

169000=132×23×53169000 = 13^2 \times 2^3 \times 5^3

Step 2: Check the exponents.
The exponents of the prime factors are 2 (for 13), 3 (for 2), and 3 (for 5). Since the exponents 3 are odd, 169000 is not a perfect square.

❌ Incorrect. 169000 is not a perfect square number.

4. Find the square root of the following using the factorization method:

Part a· 2 steps

169

  1. Find the prime factorization of 169.

    169=13×13=132169 = 13 \times 13 = 13^2

  2. Find the square root.

    169=132=13\sqrt{169} = \sqrt{13^2} = \boxed{\text{13}}

Part b

18\text{18}

324

  1. Find the prime factorization of 324.

    324=2×162324 = 2 \times 162 162=2×81162 = 2 \times 81 81=3×2781 = 3 \times 27 27=3×927 = 3 \times 9 9=3×39 = 3 \times 3

    So,

    324=2×2×3×3×3×3=22×34324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^2 \times 3^4

  2. Find the square root.

    324=22×34=222×342=21×32=2×9=18\sqrt{324} = \sqrt{2^2 \times 3^4} = 2^{\frac{2}{2}} \times 3^{\frac{4}{2}} = 2^1 \times 3^2 = 2 \times 9 = \boxed{\text{18}}

Part c

35\text{35}

1225

  1. Find the prime factorization of 1225.

    1225=5×2451225 = 5 \times 245 245=5×49245 = 5 \times 49 49=7×749 = 7 \times 7

    So,

    1225=5×5×7×7=52×721225 = 5 \times 5 \times 7 \times 7 = 5^2 \times 7^2

  2. Find the square root.

    1225=52×72=522×722=51×71=5×7=35\sqrt{1225} = \sqrt{5^2 \times 7^2} = 5^{\frac{2}{2}} \times 7^{\frac{2}{2}} = 5^1 \times 7^1 = 5 \times 7 = \boxed{\text{35}}
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