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Chapter 8: Problem 94

Simplify. All variables represent positive values. $$\sqrt{300}-\sqrt{75}$$

Short Answer

Expert verified
The simplified form of the expression \( \sqrt{300}-\sqrt{75} \) is \( 5\sqrt{3} \).

Step by step solution

01

Break Down the Square Roots

Start by factoring each of the numbers inside the square roots to its prime factors, such that among the factors there are perfect squares. For \( \sqrt{300} \), we can write \( \sqrt{100*3} \), and for \( \sqrt{75} \), we can write \( \sqrt{25*3} \).
02

Simplify the Square Roots

After finding the perfect squares, we simplify by taking the square root of these perfect squares. \( \sqrt{100*3} \) simplifies to \( 10\sqrt{3} \) because \( \sqrt{100} \) is 10. Similarly, \( \sqrt{25*3} \) simplifies to \( 5\sqrt{3} \) because \( \sqrt{25} \) is 5.
03

Subtract the Simplified Square Roots

Now with the simplified square roots, subtract \( 10\sqrt{3} \) by \( 5\sqrt{3} \). In mathematics, when you subtract two or more similar radicals, you simply subtract the coefficients of the radicals. The resultant value is \( 5\sqrt{3} \).

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Key Concepts

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

Understanding Prime Factorization
Prime factorization is a technique used to express a number as the product of its prime numbers. Prime numbers are numbers greater than 1 with no divisors other than 1 and itself. When breaking down a composite number into prime factors, we repeatedly divide by the smallest prime number until we reach 1.
For example, to factorize 300:
  • Start by dividing by the smallest prime, 2: \(300 \div 2 = 150\)
  • Divide 150 by 2: \(150 \div 2 = 75\)
  • Next, divide 75 by 3: \(75 \div 3 = 25\)
  • Finally, divide 25 by 5: \(25 \div 5 = 5\) and \(5 \div 5 = 1\)
The prime factors of 300 are: \(2^2 \times 3 \times 5^2\).
Similarly, break down 75 into prime factors:
  • Divide 75 by 3: \(75 \div 3 = 25\)
  • Divide 25 by 5: \(25 \div 5 = 5\) and \(5 \div 5 = 1\)
The prime factors of 75 are: \(3 \times 5^2\).
Using prime factorization helps us find perfect squares within numbers, which is essential for simplifying square roots.
Steps to Simplifying Radicals
The process of simplifying radicals, such as square roots, revolves around converting them into simpler forms. This is done by identifying and utilizing perfect squares within a number. A perfect square is a number which is an exact square of an integer, like 4, 9, 16, etc.
For example:
  • For \( \sqrt{300} \), we see the prime factors \(2^2 \times 3 \times 5^2\). Among these, \(100\) (which is \(10^2\)) is a perfect square. So, \(\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}\).
  • For \( \sqrt{75} \), the prime factors are \(3 \times 5^2\). Here, \(25\) (which is \(5^2\)) is a perfect square. So, \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).
By identifying these perfect squares, the square root of the perfect square is simplified, leaving behind any leftover factors as part of the radical.
Subtraction of Radicals and Final Results
Subtraction of radicals involves combining or reducing similar radical terms, focusing on their coefficients. Much like algebraic expressions, only like terms can be combined; in this case, terms with the same radical part.
Consider the simplified expressions from the example:
  • The expression \(10\sqrt{3}\)
  • The expression \(5\sqrt{3}\)
Since both terms have the radical part \(\sqrt{3}\), you subtract their coefficients:
  • \(10 - 5 = 5\)
Therefore, \(10\sqrt{3} - 5\sqrt{3} = 5\sqrt{3}\).
This demonstrates that subtracting similar radicals simply boils down to manipulating the coefficients, making the operation straightforward when radicals share the same component.

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Most popular questions from this chapter

Solve: \(\sqrt[4]{3 x+4}=5\)

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{500}{81}}$$

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{1}{64}\right)^{1 / 3}$$

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