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Chapter 5: Problem 64

Change each radical to simplest radical form. \(\frac{4 \sqrt{45}}{-6 \sqrt{20}}\)

Short Answer

Expert verified
\(-1\)

Step by step solution

01

Simplify the Radicands

To begin, simplify the radicals within the expression separately. For \(\sqrt{45}\), we find the prime factorization of 45: \(45 = 3^2 \cdot 5\). Thus, \(\sqrt{45} = \sqrt{3^2 \cdot 5} = 3\sqrt{5}\). For \(\sqrt{20}\), we find the prime factorization of 20: \(20 = 2^2 \cdot 5\). Thus, \(\sqrt{20} = \sqrt{2^2 \cdot 5} = 2\sqrt{5}\).
02

Substitute the Simplified Radicals into the Expression

Replace \(\sqrt{45}\) and \(\sqrt{20}\) with their simplified forms: \(\frac{4 \cdot 3\sqrt{5}}{-6 \cdot 2\sqrt{5}}\), which simplifies to \(\frac{12\sqrt{5}}{-12\sqrt{5}}\).
03

Simplify the Entire Expression

Next, simplify \(\frac{12\sqrt{5}}{-12\sqrt{5}}\) by recognizing that the \(\sqrt{5}\) terms cancel out, giving \(\frac{12}{-12}\), which simplifies to \(-1\).
04

Conclusion

The simplest form of the given radical expression is \(-1\).

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Key Concepts

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

Simplest Radical Form
Simplifying a radical to its simplest form involves breaking down a number within a square root to its basic components. It means expressing the radical without any perfect squares left beneath the radical sign. For example,
  • The radical \(\sqrt{45}\) can be simplified by factoring 45 into \(3^2 \cdot 5\).
  • Since \(3^2\) forms a perfect square, it can be taken out of the radical, simplifying it to \(3\sqrt{5}\).
Simplifying radicals helps in reducing the expression to its base form, making further calculations easier.
Prime Factorization
Prime factorization breaks down a number into its basic prime number components. This process is key in simplifying radicals. Every number can be expressed as a product of prime numbers, and this is particularly useful when simplifying square roots.
  • For \(\sqrt{45}\), we express 45 as \(3^2\cdot 5\).This gives us the perfect square \(3^2\).
  • Similarly, \(\sqrt{20}\) can be expressed using prime factors as \(2^2\cdot 5\).
The perfect squares \(3^2\) and \(2^2\) can then be extracted from the square root, simplifying the expression to \(3\sqrt{5}\) and \(2\sqrt{5}\) respectively, making complex expressions more manageable.
Simplifying Expressions
After simplifying the radicals within, the final step is to simplify the entire expression. This involves substituting the simplified radicals back into the original expression and reducing any like terms.
  • In our example, we substitute back \(3\sqrt{5}\) for \(\sqrt{45}\) and \(2\sqrt{5}\) for \(\sqrt{20}\):
  • The expression becomes \(\frac{4 \cdot 3\sqrt{5}}{-6 \cdot 2\sqrt{5}}\), which simplifies to \(\frac{12\sqrt{5}}{-12\sqrt{5}}\).
  • Observing that \(\sqrt{5}\) terms cancel out, we simplify the expression to \(\frac{12}{-12} = -1\).
Simplifying the entire expression not only reduces the number of components, but also helps in clarifying the solution. Ultimately, the solution becomes straightforward and easier to understand, as in this case where everything simplified to \(-1\).

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Most popular questions from this chapter

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{6 x^{\frac{2}{5}}}{7 y^{\frac{2}{3}}}\right)^{2}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{(0.00063)(960,000)}{(3,200)(0.0000021)}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{8}}{\sqrt[4]{4}}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right)\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt[3]{0.001}\)
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