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Chapter 7: Problem 22

Simplify each expression. Give exact answers. $$3 \sqrt{50}-2 \sqrt{32}$$

Short Answer

Expert verified
The simplified expression is \( 7 \sqrt{2} \).

Step by step solution

01

- Simplify each square root

First, express the radicands (the numbers inside the square roots) in terms of their prime factors: \( \sqrt{50} \) becomes \( \sqrt{25 \times 2} \) and \( \sqrt{32} \) becomes \( \sqrt{16 \times 2} \).
02

- Separate the square roots

Next, separate the square roots to simplify: \( \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \) and \( \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2} \).
03

- Multiply by the constants

Now, multiply the simplified square roots by the constants outside: \( 3 \sqrt{50} = 3 \cdot 5 \sqrt{2} = 15 \sqrt{2} \) and \( 2 \sqrt{32} = 2 \cdot 4 \sqrt{2} = 8 \sqrt{2} \).
04

- Subtract the terms

Finally, subtract the two like terms: \( 15 \sqrt{2} - 8 \sqrt{2} = 7 \sqrt{2} \).

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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 the process of breaking down a number into its basic building blocks called prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To simplify square roots, we first express the radicand (the number inside the square root) as a product of prime factors.
For example, consider the number 50. Its prime factors are 2 and 5 because 50 can be expressed as: \[ 50 = 2 \times 5^2 \] Similarly, for 32, the prime factorization is: \[ 32 = 2^5 \] Breaking down numbers into their prime factors makes it easier to simplify square roots, leading to simpler expressions.
Square Root Properties
Understanding the properties of square roots is crucial for simplifying expressions. A square root essentially finds a value that, when multiplied by itself, gives the original number. The symbols and properties we use for square roots include:
  • \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), which allows us to separate a product under one square root into the product of two square roots.
  • \( \sqrt{a^2} = a \), meaning a number squared and then square-rooted returns the original number.
In our exercise, we use these properties to simplify the square roots of 50 and 32:
For \( \sqrt{50} \), we write it as \( \sqrt{25 \times 2} \), then separate it to get \( 5 \sqrt{2} \).
Likewise, \( \sqrt{32} \) becomes \( \sqrt{16 \times 2} \), simplifying to \( 4 \sqrt{2} \).
Like Terms
In algebra, like terms are terms that contain the same variables raised to the same power. Simplifying expressions with like terms means combining these terms. When dealing with square roots, like terms will have the same radicand.
Our example involves the terms \( 3 \sqrt{50} \) and \( -2 \sqrt{32} \). After simplification, we get \( 15 \sqrt{2} \) and \( -8 \sqrt{2} \), which are like terms because they both include \( \sqrt{2} \).
To combine them, subtract the coefficients: \[ 15 \sqrt{2} - 8 \sqrt{2} = 7 \sqrt{2} \] This property helps in simplifying the expression to its final form.

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