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Chapter 8: Problem 34

Simplify. $$\sqrt{20}+\sqrt{80}-\sqrt{125}$$

Short Answer

Expert verified
The simplified result is \( \sqrt{5}\).

Step by step solution

01

Prime Factorize All Numbers Under The Square Roots

Find the prime factorisation of each number underneath the root: \(20 = 2^2 脳 5\), \(80 = 2^4 脳 5\), and \(125 = 5^3\).
02

Simplify The Square Roots

For each number under the root, bring out the multiples of 2 from under the root sign: \(\sqrt{20} = 2\sqrt{5}\), \(\sqrt{80} = 2^2\sqrt{5} = 4\sqrt{5}\), and \(\sqrt{125} = 5\sqrt{5}\).
03

Perform the Addition and Subtraction of Square Roots

Now, add the simplified square roots and subtract the simplified square roots: \(2\sqrt{5} + 4\sqrt{5} - 5\sqrt{5}\).
04

Combine Like Terms

Combine the like terms to get the final answer: \( (2 + 4 - 5)\sqrt{5} = \sqrt{5}\).

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Key Concepts

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

Prime Factorization
Prime factorization involves breaking down a number into its prime numbers, which are numbers only divisible by 1 and themselves. For instance, when you are given a number like 20, the process involves finding the smallest prime number that divides it perfectly. The prime factors of 20 are 2 and 5 since 20 can be broken down into \( 2^2 \times 5 \). This helps in simplifying square roots.
  • Identify the smallest prime number that divides the number.
  • Divide the entire number by this prime number.
  • Repeat the process with the quotient until only prime numbers are left.
Prime factorization is handy in simplifying radical expressions because it allows you to easily group numbers to pull them out from under a square root.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). With prime factorization, simplifying square roots becomes easier.
  • Simplification: Once you have the prime factorization, you can simplify the square root by "bringing out" pairs of prime numbers. Each pair represents a perfect square that can be taken outside the square root.
  • Example: In \( \sqrt{80} \), the prime factorization is \( 2^4 \times 5 \). You can take \( 2^2 \) out of the square root because \( 4 \) is a perfect square: \( 4 \sqrt{5} \).
Knowing how to simplify square roots using prime factorization is crucial in dealing with radical expressions efficiently.
Combining Like Terms
Combining like terms is a fundamental algebraic process where similar terms are added or subtracted to simplify expressions. This concept is essential when working with radicals, especially after simplifying them.
  • Like Terms: These are terms that have the same variable raised to the same power, such as \( 2 \sqrt{5} \) and \( 4 \sqrt{5} \). Only like terms can be combined.
  • Combining: This involves adding or subtracting the coefficients (the numbers in front of the radicals) while keeping the radical unchanged. In \( 2\sqrt{5} + 4\sqrt{5} - 5\sqrt{5} \), the terms are combined as follows: \((2+4-5)\sqrt{5} \).
  • Final Result: The expression simplifies to \( \sqrt{5} \), as the addition and subtraction of the coefficients \( (2+4-5) = 1 \) lead to this final term.
Mastering the technique of combining like terms makes it easier to handle complex radical expressions and simplify them accurately.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{4}{49}\right)^{1 / 2}$$

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Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{432}$$

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