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Chapter 9: Problem 96

Simplify. $$ 3+\sqrt{125} $$

Short Answer

Expert verified
3 + 5\sqrt{5}

Step by step solution

01

Identify the Square Root

Recognize that \(\sqrt{125}\) can be simplified by finding the prime factors of 125.
02

Factor 125

Since 125 can be factored into \(5 \times 5 \times 5\), rewrite it as \(\sqrt{5^3}\).\
03

Simplify the Square Root

Use the property of square roots \(\sqrt{a^2b} = a \sqrt{b}\) to simplify \(\sqrt{5^3}\) as \(5\sqrt{5}\).\
04

Combine Terms

Now add the simplified square root to the constant: \(3 + 5\sqrt{5}\).

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Key Concepts

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

Square Roots
Square roots are numbers that produce a specified number when multiplied by themselves. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. The symbol for square root is \( \sqrt{...} \). Determining square roots requires an understanding of multiplication and, in some cases, prime factorization.
Prime Factorization
Prime factorization involves breaking a number down into its prime components. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves. For 125, we check its prime factors as follows: \ \[ 125 = 5 \times 25 = 5 \times 5 \times 5 = 5^3 \ \]. This step is crucial for simplifying square roots effectively.
Mathematical Properties
Mathematical properties help in manipulating and simplifying expressions. One key property for square roots is: \[ \sqrt{a^2 b} = a \sqrt{b} \ \]. Applying this property, \( \sqrt{5^2 \times 5} = 5 \sqrt{5} \). Understanding this property is essential to simplify the expression effectively.
Simplified Form
The simplified form of mathematical expressions makes them easier to understand and use. To simplify \ \( 3 + \sqrt{125} \):
  • First, find the prime factors of 125 to get \( \sqrt{5^3} \).
  • Then simplify it to get \ 5 \sqrt{5} \ \.
  • Finally, combine it with 3 to get \ 3 + 5 \sqrt{5} \ \.
This is the cleanest and most manageable form.

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Most popular questions from this chapter

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