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Chapter 8: Problem 19

Simplify each radical. $$ \sqrt{90} $$

Short Answer

Expert verified
The simplified form of \( \sqrt{90} \) is 3\( \sqrt{10} \).

Step by step solution

01

Find Prime Factors

Determine the prime factorization of 90. Start by dividing 90 by the smallest prime numbers until only prime numbers are left.
02

Prime Factorization

Prime factorization of 90 is: 90 = 2 脳 3 脳 3 脳 5.
03

Grouping Prime Factors

Identify pairs of the same prime numbers. In this case, we have a pair of 3's.
04

Simplify the Radical

Rewrite the square root including the pair of 3's: o \( \sqrt{90} = \sqrt{(3 \times 3) \times 2 \times 5} \).Extract the pair outside the radical as a single number: \( \sqrt{90} = 3 \ times\sqrt{(2 \times 5)} \ = 3 \sqrt{10} \).

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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 a method to express a number as the product of its prime numbers. For any composite number, it can be broken down into prime numbers, which are numbers greater than 1 and only divisible by 1 and themselves. Here鈥檚 how to perform prime factorization:
  • Start with the smallest prime number, which is 2, and check if it divides the number without leaving a remainder.
  • If it divides the number, then factor it out and repeat the process with the quotient.
  • If the current smallest prime doesn't divide the number, move to the next smallest prime.
For example, to find the prime factors of 90, we would divide by 2 (since 90 is even), giving us 45. Then we divide 45 by 3, giving us 15, and divide 15 by 3 again, giving us 5. All these numbers (2, 3, 3, 5) are prime, so 90 = 2 脳 3 脳 3 脳 5.
Square Roots
A square root of a number is a value which, when multiplied by itself, gives the original number. The radical sign \( \sqrt{} \) is used to denote a square root. For instance, the square root of 16 is 4, because 4 脳 4 = 16.
  • When simplifying square roots, it is helpful to know the factors of the number inside the radical sign.
  • You will often use prime factorization to find pairs of prime factors, since squaring and then taking the square root will bring a pair out of the radical as a single number.
For example, with \( \sqrt{90} \), we can express it as \( \sqrt{(3 \times 3) \times 2 \times 5} \). Given a pair of 3鈥檚, one 3 can be taken out of the radical, leaving \( 3 \sqrt{10} \).
Mathematical Simplification
Mathematical simplification involves reducing an expression to its simplest form. This process makes expressions easy to read and understand. For simplifying radicals, follow these steps:
  • Factorize the number under the radical into its prime factors.
  • Group the prime factors and pull out pairs from the radical.
  • Multiply the numbers outside the radical for your final answer.
Let鈥檚 simplify \( \sqrt{90} \):
  • Prime factorize 90: 90 = 2 脳 3 脳 3 脳 5.
  • Group the factors: \( \sqrt{(3 \times 3) \times 2 \times 5} \).
  • Bring out one number from each pair: 3\( \sqrt{10} \).
So, \( \sqrt{90} \) simplifies to 3 \( \sqrt{10} \). Simplification helps in solving equations more efficiently and understanding the relationships between different components clearly.

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Most popular questions from this chapter

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