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Chapter 5: Problem 36

Change each radical to simplest radical form. \(\frac{3}{4} \sqrt{45}\)

Short Answer

Expert verified
\( \frac{9}{4} \sqrt{5} \)

Step by step solution

01

Factor Inside the Radical

Firstly, factor the number inside the square root, \( \sqrt{45} \). The prime factorization of 45 is \( 3^2 \times 5 \). Therefore, \( \sqrt{45} = \sqrt{3^2 \times 5} \).
02

Simplify the Radical

Apply the property of square roots that allows you to separate the factors: \( \sqrt{3^2 \times 5} = \sqrt{3^2} \times \sqrt{5} \). \( \sqrt{3^2} \) simplifies to 3, so \( \sqrt{45} = 3 \sqrt{5} \).
03

Simplify the Expression

Now we have the expression \( \frac{3}{4} \times 3 \sqrt{5} \). First, multiply the fractions: \( \frac{3}{4} \times 3 = \frac{9}{4} \). So the expression simplifies to \( \frac{9}{4} \sqrt{5} \).

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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 a technique used to express a number as the product of its prime numbers. It plays a crucial role when dealing with radicals. To solve square roots, we need to break down the number under the radical sign to its basic components. This is known as prime factorization.

For example, in the original exercise with the number 45, we determine its prime factorization by continuously dividing it by the smallest prime numbers.
  • First, divide 45 by 3, a prime number: this gives 15.
  • Next, divide 15 by 3 again, which gives 5, another prime number.
So, the prime factors of 45 are 3 and 5, and they can be expressed as \(3^2 \times 5\). This expression helps pave the way for further simplification using the square root property.
Square Root Property
The square root property is integral in simplifying numbers under the radical sign. It allows us to break apart numbers into their prime factors and simplifies any perfect squares.

Taking 45 as our example again, we use its prime factorization: \(3^2 \times 5\). Here's how the square root property aids in simplification:
  • Apply the square root to each part of the factorization: \( \sqrt{3^2 \times 5} \).
  • Use the property that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the terms inside: this becomes \( \sqrt{3^2} \times \sqrt{5} \).
Since \( \sqrt{3^2} = 3 \), the expression under the radical simplifies to \(3 \sqrt{5}\). The square root property makes it possible to identify and extract perfect squares from under the radical, leading to a simpler expression.
Simplifying Expressions
Once we have a simplified radical, the next step is to simplify the entire expression. Simplifying expressions requires both arithmetic manipulation of fractions and radicals.

In our example, we started with \( \frac{3}{4} \sqrt{45} \). After simplifying the radical to \(3 \sqrt{5}\), the expression turned into \( \frac{3}{4} \times 3 \sqrt{5} \). Here are the steps to simplify this further:
  • Multiply the fraction \( \frac{3}{4} \) by 3, resulting in \( \frac{9}{4} \).
  • This yields the simplified form: \( \frac{9}{4} \sqrt{5} \).
Breaking down expressions into their simplest forms isn't just about making them look neat. It鈥檚 about making calculations easier and ensuring that the expression communicates the most information in the most efficient way possible.

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Most popular questions from this chapter

The field of view of a microscope is \((4)\left(10^{-4}\right)\) meters. If a single cell organism occupies \(\frac{1}{5}\) of the field of view, find the length of the organism in meters. Express the result in scientific notation.

A square pixel on a computer screen has a side of length (1.17) \(\left(10^{-2}\right)\) inches. Find the approximate area of the pixel in inches. Express the result in decimal form.

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (5)\left(10^{8}\right) $$

For Problems \(81-90\), perform the indicated operations and express answers in simplest radical form. $$ \sqrt[3]{3} \sqrt{3} $$

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{a^{3}}{b^{-2}}\right)^{-\frac{1}{3}} $$
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