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Chapter 10: Problem 66

\(\sqrt{1500}\)

Short Answer

Expert verified
\( 10 \sqrt{15} \)

Step by step solution

01

Simplify the Expression

First, write the expression in simpler terms by factoring 1500 into its prime factors. Notice that 1500 can be written as 15 * 100.
02

Break Down the Factors

Factorize 15 and 100 further into their prime factors. We get: 15 = 3 * 5 and 100 = 2^2 * 5^2.
03

Combine Prime Factors

Combine all the prime factors of 1500. Therefore, 1500 = 3 * 5 * 2^2 * 5^2, which simplifies to 1500 = 2^2 * 3 * 5^3.
04

Apply the Square Root

Apply the square root to the simplified factorization: \ \ \ \ \ \ \ \ \ \ \(\sqrt{1500} = \sqrt{2^2 * 3 * 5^3}\) \ \ \ \ \ \ This breaks down to \(\sqrt{2^2} * \sqrt{3} * \sqrt{5^3}\).
05

Simplify Further

Evaluate each square root separately: \(\sqrt{2^2} = 2\), \(\sqrt{3} \), and \(\sqrt{5^3} = 5 * \sqrt{5}\). Combining these, we get: \(\sqrt{1500} = 2 * \sqrt{3} * 5* \sqrt{5} \).
06

Final Simplification

Combine the results from the previous step: \(\sqrt{1500} = 2*5 \sqrt{3*5} = 10 \sqrt{15}\).

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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 the process of expressing a number as the product of prime numbers. For instance, we can break down 1500 into its prime factors: 15 and 100. Then, further decompose these:
  • 15 = 3 * 5 (since 3 and 5 are prime)
  • 100 = 2^2 * 5^2 (since 2 and 5 are prime)
Combining these, we get the prime factorization of 1500:
  • 1500 = 2^2 * 3 * 5^3
Prime factorization helps simplify complex mathematical expressions, making it easier to perform operations like square roots. By decomposing numbers into primes, we can systematically find solutions.
Simplifying Radicals
Simplifying radicals involves breaking down a square root into its simplest form. Let's simplify \(\root(1500)\). First, use prime factorization: \(\root(2^2 * 3 * 5^3)\). Then, separate terms where possible: \(\root(2^2) * \root(3) * \root(5^3)\).
  • \root(2^2) = 2
  • \root(3)= \root(3)
  • \root(5^3) = 5 * \root(5)
Combine the simplified roots: 2 * 5 \root(3*5) = 10 \root(15). This method simplifies complex radicals into more manageable forms.
Algebraic Expressions
Algebraic expressions include variables, constants, and operations. When simplifying square roots within these expressions, follow systematic steps. For example: \(\root(2^2 * 3 * 5^3)\) simplifies to \(\root(2^2) * \root(3) * \root(5^3)\), which becomes 2 * \root(3) 5 \root(5). Combining these, we get 10 \root(15). Algebra simplifies radicals and other expressions logically. Follow rules and breakdown steps efficiently to deal with complex problems in mathematics.

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