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Chapter 5: Problem 24

Simplify. $$ \sqrt{210,000} $$

Short Answer

Expert verified
100 \sqrt{21}

Step by step solution

01

- Prime Factorization

Start by finding the prime factorization of 210,000. We have:\[ 210,000 = 21 \times 10,000 \]Now factorize each component:\[ 21 = 3 \times 7 \]\[ 10,000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4 \]So, the prime factorization is:\[ 210,000 = 2^4 \times 5^4 \times 3 \times 7 \]
02

- Write Under the Square Root

Now substitute the prime factors back under the square root:\[ \sqrt{210,000} = \sqrt{2^4 \times 5^4 \times 3 \times 7} \]
03

- Simplify the Radicand

Use the property of square roots that \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}:\[ \sqrt{210,000} = \sqrt{2^4} \times \sqrt{5^4} \times \sqrt{3} \times \sqrt{7} \]Since \sqrt{2^4} = 2^2 = 4 and \sqrt{5^4} = 5^2 = 25, it simplifies to:\[ 4 \times 25 \times \sqrt{3 \times 7} = 100 \times \sqrt{21} \]

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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 used to break down a number into its smallest prime components. This means expressing the number as a product of prime numbers, which are numbers greater than 1 that have no divisors other than 1 and themselves. For example, in our exercise to simplify \(\sqrt{210,000}\):
  • First, we broke 210,000 into smaller factors: \(210,000 = 21 \times 10,000\)
  • Next, we factorized each part into primes: \(21 = 3 \times 7\) and \(10,000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4\).
Putting it all together, we get the prime factorization: \[ 210,000 = 2^4 \times 5^4 \times 3 \times 7 \]. This breakdown makes it much easier to work with square roots later.
Properties of Square Roots
Understanding the properties of square roots helps in simplifying complex square root problems. The key property used in our exercise is the multiplication rule for square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This rule allows us to split the square root of a product into the product of individual square roots.
For \(\sqrt{210,000}\), we substitute the prime factors back under the square root: \[ \sqrt{210,000} = \sqrt{2^4 \times 5^4 \times 3 \times 7} \]. This property simplifies our task as we can now handle each prime factor separately.
Radical Simplification
Radical simplification is the process of simplifying square roots to their simplest form. By using the properties of square roots, we can simplify even complicated expressions. For \(\sqrt{210,000}\), we already factored it into \(2^4 \times 5^4 \times 3 \times 7\).
The next step is to apply the square root to each factor:
  • \(\sqrt{2^4} = 2^2 = 4\)
  • \(\sqrt{5^4} = 5^2 = 25\)
Now, multiply the results and keep the remaining terms under the square root: \[ 4 \times 25 \times \sqrt{3 \times 7} = 100 \times \sqrt{21} \].
This method breaks down the problem into smaller, manageable pieces, making complex square roots easier to understand and simplify.

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Most popular questions from this chapter

Find the value of c in each equation. $$\sqrt[c]{81}=3$$

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Find the value of \(t\) in each equation. \(t^{5}=32\)
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