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Chapter 10: Problem 25

Perform the operation and simplify. Assume all variables represent non negative real numbers. $$\sqrt{32}-3 \sqrt{8}$$

Short Answer

Expert verified
The simplified expression is \(-2\sqrt{2}\).

Step by step solution

01

Simplify the first square root

To simplify the square root of 32, we'll find the prime factorization of 32: \(32 = 2^5\). Since there are 5 factors of 2, we can take out two pairs of 2 from the square root: \[\sqrt{32} = \sqrt{2^5} = 2^2\sqrt{2} = 4\sqrt{2}\]
02

Simplify the second square root

To simplify the square root of 8, we'll find the prime factorization of 8: \(8 = 2^3\). Since there are 3 factors of 2, we can take out one pair of 2 from the square root: \[\sqrt{8} = \sqrt{2^3} = 2\sqrt{2}\]
03

Perform the subtraction

Now that we have simplified the square roots, we can perform the subtraction: \(4\sqrt{2} - 3\sqrt{8} = 4\sqrt{2} - 3(2\sqrt{2})\) Since both terms have a common factor of \(\sqrt{2}\), we can further simplify: \(4\sqrt{2} - 6\sqrt{2} = (4-6)\sqrt{2} = -2\sqrt{2}\)
04

Final Answer

The simplified expression is \(-2\sqrt{2}\).

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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 a fundamental concept in mathematics, representing a number that, when multiplied by itself, results in the original number. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4. Square roots are symbolized by the radical sign \(\sqrt{}\).
Understanding square roots is essential for simplifying expressions, like in the given problem: \(\sqrt{32}-3 \sqrt{8}\). Here, the objective is to simplify each square root before performing any operations.
When simplifying square roots,
  • Identify perfect squares within the number.
  • Factorize the number to find these perfect squares.
  • Extract the square root of perfect squares so they sit outside the radical.
By doing this, we transform complex square roots into simpler forms to easily work with them in algebraic operations.
Prime Factorization
Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is any number greater than 1 that has no positive divisors other than 1 and itself. Understanding prime factorization is crucial for simplifying radical expressions.
When factoring a number, like 32 in the first step of the original solution, the goal is to express it as a product of prime numbers:
  • For 32, the prime factorization is \(2^5\) because 32 can be written as \(2 \times 2 \times 2 \times 2 \times 2\).
  • For 8, the prime factorization is \(2^3\) as 8 is \(2 \times 2 \times 2\).
This factorization makes it easy to simplify square roots by grouping pairs of identical prime factors, allowing us to extract them from under the radical sign efficiently.
Algebraic Operations
Algebraic operations involve manipulating expressions using addition, subtraction, multiplication, and division. When dealing with radical expressions, proper simplification through prime factorization and recognizing common factors play a vital role in performing these operations.
In expressions like \(4\sqrt{2} - 3\sqrt{8}\), simplifying each term first allows us to combine them effectively:
  • Simplify each square root individually by using prime factorization.
  • Identify common terms or factors, like \(\sqrt{2}\) in this example.
  • Perform operations such as subtraction or addition only on the coefficients of the radicals.
For the subtraction operation in this problem, extracting the factor \(\sqrt{2}\) from both terms made it easier to solve: \(4\sqrt{2} - 6\sqrt{2} = (4-6)\sqrt{2} = -2\sqrt{2}\). This approach ensures clarity and accuracy in solving radical expressions.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[5]{\frac{3}{8}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{b-25}{\sqrt{b}-5}$$

Rationalize the denominator of each expression. $$\sqrt{\frac{42}{35}}$$

Find each root, if possible. $$\sqrt{5^{2}+12^{2}}$$

In your own words, explain how to rationalize the denominator of an expression containing one term in the denominator.
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