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Chapter 7: Problem 55

Simplify. \(\sqrt{30}\)

Short Answer

Expert verified
鈭�30 cannot be simplified further.

Step by step solution

01

Identify Perfect Square Factors

Check if the number inside the square root, 30, has any perfect square factors. In this case, it does not, as 30 is the product of 2, 3, and 5, none of which are perfect squares.
02

Simplify Further

Since 30 has no perfect square factors, \(\text{\textbackslash sqrt\textbraceleft 30\textbraceright}\) cannot be simplified further.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

perfect square factors
Perfect square factors are numbers that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be written as 2\textsuperscript{2}, and 9 is a perfect square because it can be written as 3\textsuperscript{2}. Identifying these factors helps in simplifying square roots.
If a number has perfect square factors, we can break it down to make the square root simpler. However, for \( \sqrt{30} \), there are no perfect square factors since the number 30 is the product of 2, 3, and 5, which are all prime numbers.
When we cannot find any perfect square factors, it means we cannot simplify the square root any further.
radical expressions
Radical expressions include any expression that contains a square root, cube root, or other roots. The symbol used is called the radical sign ( \( \sqrt{} \) ).
In this case, \( \sqrt{30} \) is a radical expression with 30 under the radical sign. To simplify a radical expression, we look for factors of the number inside the radical that are perfect squares.
For instance, \( \sqrt{8} = \sqrt{4 \times 2} \), because 4 is a perfect square. This simplifies to \( 2 \sqrt{2} \). However, for \( \sqrt{30} \), because there are no perfect square factors, it stays as it is.
prime factorization
Prime factorization is breaking down a number into its prime number factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
To find the prime factorization of 30, we divide it by the smallest prime number that can go into it evenly, repeatedly:
  • 30 divided by 2 (the smallest prime) is 15
  • 15 divided by 3 (next prime) is 5
  • 5 is itself a prime number
    So the prime factors of 30 are 2, 3, and 5. In our case, since these prime factors themselves aren't perfect squares, we see there's no further simplification possible for \( \sqrt{30} \).

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