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Solution - Square root of fraction or number by prime factorization

(s q r t (30)) / 600

(sqrt(30))/600

Decimal form:

0.009

0.009

See steps

Step-by-step explanation

1. Reduce the fraction to its lowest terms

Divide both the numerator and denominator by their greatest common factor (1):

Since the GCF is 1, the fraction cannot be reduced $\sqrt{\frac{1}{12000}}$

Learn how to find the greatest common factor.

2. Find the prime factors of 1

1 is a prime factor.

$1 = 1$

3. Find the prime factors of 12,000

Tree view of the prime factors of 12,000: 2, 2, 2, 2, 2, 3, 5, 5 and 5

The prime factors of 12,000 are 2, 2, 2, 2, 2, 3, 5, 5 and 5.

$12000 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 5$
$12000 = 2^{5} \cdot 3 \cdot 5^{3}$

4. Express the fraction in terms of its prime factors

$\sqrt{\frac{1}{12000}} = \frac{\sqrt{1}}{\sqrt{12000}}$

Write the prime factors:

$s q r t ((1)) / s q r t ((12000)) = (1) / s q r t (2 * 2 * 2 * 2 * 2 * 3 * 5 * 5 * 5)$

Group the prime factors into pairs and rewrite them in exponent form:

$(1) / s q r t (2 * 2 * 2 * 2 * 2 * 3 * 5 * 5 * 5) = (1) / s q r t (2^{2} * 2^{2} * 2 * 3 * 5^{2} * 5)$

Use the rule $\sqrt{(x^{2})} = x$ to simplify further:

$(1) / s q r t (2^{2} * 2^{2} * 2 * 3 * 5^{2} * 5) = (1) / (2 * 2 * 5 * s q r t (2 * 3 * 5))$

Perform any multiplication or division, from left to right:

$(1) / (2 * 2 * 5 * s q r t (2 * 3 * 5)) = (1) / (4 * 5 * s q r t (2 * 3 * 5))$

$(1) / (4 * 5 * s q r t (2 * 3 * 5)) = (1) / (20 * s q r t (2 * 3 * 5))$

Perform any multiplication or division, from left to right:

$(1) / (20 * s q r t (2 * 3 * 5)) = (1) / (20 * s q r t (6 * 5))$

$(1) / (20 * s q r t (6 * 5)) = (1) / (20 * s q r t (30))$

Rationalize the denominator by multiplying both the numerator and denominator by the square root found in the denominator:

$(1) / (20 * s q r t (30)) = (1 * s q r t (30)) / (20 * s q r t (30) * s q r t (30))$

$(1 * s q r t (30)) / (20 * s q r t (30) * s q r t (30)) = (1 * s q r t (30)) / (20 * 30)$

$(1 * s q r t (30)) / (20 * 30) = (1 * s q r t (30)) / (600)$

$(1 * s q r t (30)) / 600 = (s q r t (30)) / 600$

The square root of $s q r t (1 / 12000)$ is $(s q r t (30)) / 600$

Decimal form: $0.009$

The principal square root is the positive number that is derived from solving a square root. For example, the principal square root of $\sqrt{(4)}$ is $2$ , $(\sqrt{(4)} = 2)$ .
$- 2$ is also a square root of $4$ , $(- 2^{2} = 4)$ , but, because it is negative, it is not the principal square root. In order to find the square of $- 2$ we need to write the equation as $- \sqrt{(4)} = - 2$ .

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Why learn this

The key to understanding and solving complex math problems is building up a wide knowledge of simpler concepts that all build on each other. One of these concepts is finding the square root of numbers or fractions using prime factorization. While this concept is important for understanding other concepts in math - for example, the Pythagorean theorem - finding square roots has many real-world applications. These include, but are not limited to, creating powerful algorithms that can solve complex problems and tackling tough engineering or architectural challenges. Prime factorization is simply a way of calculating large square roots more easily using their prime number factors.

Terms and topics

Square root of fraction or number by prime factorization