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Chapter 8: Problem 25

Simplify each radical. $$ \sqrt{145} $$

Short Answer

Expert verified
\( \sqrt{145} \) cannot be simplified further.

Step by step solution

01

Recognize the Radical

Identify the radical expression that needs to be simplified, which is \( \sqrt{145} \).
02

Prime Factorize the Radicand

Prime factorize 145 to find if there are any factor pairs. The prime factorization of 145 is 5 脳 29. Neither 5 nor 29 is a perfect square.
03

Simplify if Possible

Since neither 5 nor 29 is a perfect square, the radical \( \sqrt{145} \) cannot be simplified further.

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Key Concepts

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

Prime Factorization
Prime factorization is breaking down a number into a product of its prime numbers. This helps in simplifying radicals. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, consider the number 145. To factorize it, we find that 145 can be divided by 5, giving us 29.
So, the prime factorization of 145 is: \[145 = 5 \times 29\]Neither 5 nor 29 have other prime factors. Prime factorization is crucial because it's the basis for simplifying radicals by identifying perfect squares.
Radicals
Radicals represent the root of a number. The most common radical is the square root, but there are also cube roots, fourth roots, and so on.
The symbol for radicals is \( \sqrt{} \). When there is no small number indicating a specific root (like 3 for cube root), it's assumed to be a square root.
For instance, \( \sqrt{145} \) is asking for the square root of 145. In the context of simplifying radicals, we look for factors that are perfect squares. If we find a perfect square, we can simplify the radical. If not, like in the case of \( \sqrt{145} \), the radical remains in its original form.
Square Roots
Square roots deal with finding a number that, when multiplied by itself, gives the original number. For example, \( \sqrt{36} = 6 \), because \( 6 \times 6 = 36 \).
Not all numbers have an exact square root. For non-perfect squares like 145, finding the square root results in a decimal. However, if the number can be factored into perfect squares, it can be simplified.
In our example \( \sqrt{145} \), since 145 factors into \( 5 \times 29 \) and neither factor is a perfect square, the radical cannot be further simplified.
Understanding square roots is important in simplifying radicals, as it lets you know when a radical is already in its simplest form.

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