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Chapter 7: Problem 28

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$\sqrt{300}$$

Short Answer

Expert verified
\10\sqrt{3} \,

Step by step solution

01

- Prime Factorization

Break down 300 into its prime factors. 300 can be written as: \[300 = 2^2 \times 3 \times 5^2\]
02

- Apply the Square Root to Each Factor

Use the property of square roots that allows us to take the square root of each factor individually: \[\text{Let} \, x = 300 = 2^2 \times 3 \times 5^2 \ \text{Then}, \, \ \ \ \ \ \ \ ow sqrt{x} = \ \ \sqrt{2^2 \times 3 \times 5^2} = \sqrt{2^2} \times \sqrt{3} \times \sqrt{5^2}\]
03

- Simplify the Square Roots

Take the square root of the perfect square factors: \[\sqrt{2^2} = 2 \, \text{and} \, \sqrt{5^2} = 5 \ \text{So}, \, \ = 2 * 5 * sqrt{3} \ent * sqrt{3}\]
04

- Final Simplification

Multiply the results we have: 2 times 5 gives us 10: \[10\sqrt{3}\]

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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 basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Examples of prime numbers are 2, 3, 5, 7, and 11.
To perform prime factorization on a number, follow these steps:
  • Start with the smallest prime number, which is 2. Check if 2 divides the number evenly.
  • If it does, then divide the number by 2 and continue dividing the result by 2 until it no longer divides evenly.
  • Move to the next smallest prime number, which is 3, and repeat the process.
  • Continue this process with each successive prime number.
For example, the prime factorization of 300 is: \ \ \[ 300 = 2^2 \times 3 \times 5^2 \] Breaking 300 down, we see that:
  • 300 梅 2 = 150
  • 150 梅 2 = 75
  • 75 梅 3 = 25
  • 25 梅 5 = 5
  • 5 梅 5 = 1
Therefore, 300 can be written as the product of its prime factors: \(2^2 \times 3 \times 5^2\).
Properties of Square Roots
Square roots are mathematical operations used to find a number which, when multiplied by itself, produces the original number.
Some important properties of square roots include:
  • The square root of a product: \( \ \sqrt{a \times b} = \ \sqrt{a} \times \ \sqrt{b} \)
  • The square root of a fraction: \( \ \sqrt{\frac{a}{b}} = \ \frac{\sqrt{a}}{\sqrt{b}} \)
  • The square of a square root: \( \ \sqrt{a}^2 = a \)
  • Non-negative values: All square roots of non-negative numbers are non-negative.
Using these properties, we can simplify the square root of a number. For example, in the exercise provided, we use the square root properties to simplify \( \ \sqrt{300} \)
We apply the properties to split and simplify: \(\ \sqrt{2^2 \times 3 \times 5^2} = \const text_blank simpleheceba{} = \ \sqrt{2^2} \times \ \sqrt{3} \times \ \sqrt{5^2}\).
Finally simplifying the perfect squares, \(\ \sqrt{2^2} = 2\) and \(\ \sqrt{5^2} = 5\), we get \(2 \times 5 \times \ \sqrt{3} = 10\ \sqrt{3}\).
Perfect Squares
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, and 25 are all perfect squares because:
  • \( 1 = 1 \times 1 \)
  • \( 4 = 2 \times 2 \)
  • \( 9 = 3 \times 3 \)
  • \( 16 = 4 \times 4 \)
  • \( 25 = 5 \times 5 \)
Recognizing perfect squares is useful for simplifying square roots, as the square root of a perfect square is always an integer.
As seen in the exercise, the factors \(2^2\) and \(5^2\) are perfect squares. Hence, \(\ \sqrt{2^2} = 2\) and \(\ \sqrt{5^2} = 5\).
Only the non-perfect square factor \(3\) remains under the square root sign, resulting in the simplified form \(10\ \sqrt{3}\).

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Most popular questions from this chapter

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[10]{a} \sqrt[5]{a^{2}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[4]{a^{2} b}(\sqrt[3]{a^{2} b}-\sqrt[5]{a^{2} b^{2}}) $$

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Find the midpoint of the segment with the given endpoints. $$ (\sqrt{2},-1) \text { and }(\sqrt{3}, 4) $$
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