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Chapter 8: Problem 63

Simplify each radical expression. $$ \frac{\sqrt{75}}{\sqrt{3}} $$

Short Answer

Expert verified
\(\frac{\sqrt{75}}{\sqrt{3}} = 5\)

Step by step solution

01

Simplify the radicands

First, simplify the radicands inside of the radicals. In this problem, we have \ \(\frac{\root3{75}}{\root3{3}}\). Notice that \ \(\root3{75} = \root3{25 \times 3}\).
02

Apply the Quotient Rule for Radicals

According to the Quotient Rule for Radicals, we can combine the radicals under a single radical symbol: \ \ \(\frac{\root3{25 \times 3}}{\root3{3}} = \root3{\frac{75}{3}}\).
03

Simplify the Division

Next, simplify the expression under the root. Since \(\frac{\root3{75}}{\root3{3}} = \root3{\frac{75}{3}}\), then divide: \ \(75 \div 3 = 25\). Therefore, \ \ \(\frac{\root3{75}} = \root3{25}\).
04

Take the Square Root

Finally, take the square root of the simplified radicand: \ \(\root2{25} = 5\). Therefore, the simplified expression is: \ \ \(\frac{\root3{75}}{\root3{3}} = 5\).

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Key Concepts

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

radicands
When simplifying radical expressions, the first step often involves simplifying the 'radicand.' The radicand is the number or expression inside the radical symbol. In our original exercise, the radicands are 75 and 3 inside their respective square roots.

To simplify radicands, you need to break them down into their prime factors or recognize common factors. For example, 75 can be factored into 25 and 3, which are easier to work with inside the radical.

This step helps in preparing the expression for further simplification. Always look for factors that can be simplified or canceled out when inside a radical.
quotient rule for radicals
A key concept for simplifying radical expressions is the Quotient Rule for Radicals. This rule allows you to combine the division of two radicals into a single radical and vice versa. Mathematically, the rule is: \(\frac{\root{a}{x}}{\root{a}{y}} = \root{a}{\frac{x}{y}}\).

In our exercise, we start with \(\frac{\root{2}{75}}{\root{2}{3}}\). Using the Quotient Rule for Radicals, this simplifies to \(\root{2}{\frac{75}{3}}\), making our work easier by combining the two radicals into one.

This rule is extremely useful for simplifying complex expressions and should be applied whenever you encounter a division of radicals.
square root
The square root is a common type of radical symbol and denotes the number that, when multiplied by itself, gives the radicand. Simplifying square roots is an essential skill in algebra.

In our example, after applying the Quotient Rule for Radicals, we are left with \(\root{2}{25}\), which is simplified by recognizing that 25 is a perfect square. Therefore, \(\root{2}{25} = 5\).

When simplifying square roots:

  • Look for perfect squares within the radicand.

  • Break down the radicand into smaller factors.

  • Use the property that \(\root{2}{a^2} = a\), where a is any positive number.

Practice simplifying different square roots to get comfortable with it.

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Most popular questions from this chapter

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Find each product and simplify. $$ \sqrt{9} \cdot \sqrt{32} $$

Simplify each radical. $$ \sqrt[3]{-\frac{1}{64}} $$
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