Problem 132 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{45 r^{3}}{s^{10}}} $$

Step-by-Step Solution

Answer

\( \frac{3 \sqrt{5} r^{3/2}}{s^5} \).

Step 1 - Simplify the Radicand

Simplify the expression inside the square root. The given expression is \(\frac{45 r^{3}}{s^{10}}\). Begin by factoring 45 into prime factors, which gives 45 as 3^2 * 5.

Step 2 - Separate the Square Root

Use the property of square roots to separate the radicand into the product of square roots: \(\frac{45 r^{3}}{s^{10}} = \frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\frac{\frac{\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{\frac{\frac{\frac{\frac{\frac{\frac}}}{s^{10}}}\).

Step 3 - Simplify Roots

Simplify each part separately. For the numerator: \(\frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{3^{2} \cdot 5}{r^{3}}}{s^{10}} = \frac{\underline{\phantom{xxx}}\underline{\phantom{xxx}}\underline{\phantom{xxx}}\frac{3^{2}}{r^{2}}}{s^{10}} \). Now simplify using the rule, \(\frac{\underline{\phantom{xxx}}\frac{3^{2}}{r^{2}}}{s^{10}}\).

Step 4 - Combine Results

Combine the simplified results from the numerator and the denominator: \( \sqrt{\frac{45 r^{3}}{s^{10}}} = \frac{3 \sqrt{5} r^{3/2}}{s^5} \).

Key Concepts

Radicand

The radicand is the number or expression inside the square root symbol. In the given exercise, the radicand is \(\frac{45 r^{3}}{s^{10}}\). Understanding the radicand helps in simplifying the square root expression. We need to break down what's inside the radical to proceed with simplification. Start by factoring any composite numbers and simplifying the contents.

Prime Factorization

Prime factorization involves breaking down a number into the product of its prime factors. For the number 45, the prime factorization is \(3^2 \times 5\). Prime factorization helps us to simplify the square root more easily by dealing with smaller prime units. For example, writing 45 as \(3^2 \times 5\) allows us to see which parts can be taken out of the square root.

Properties of Square Roots

Square roots have certain properties that are useful in simplification:

• \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\): You can split the square root of a product into the product of the square roots.
• \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\): The square root of a fraction is the fraction of the square roots.
• Recognizing these properties helps us break down and simplify more complex expressions.

Simplifying Fractions

When simplifying square roots that include fractions, follow a systematic approach:

• Simplify the numerator and the denominator separately.
• Use prime factorization where applicable.
• Apply properties of square roots to deal with each component.
  For example, in the problem \(\sqrt{\frac{45 r^{3}}{s^{10}}}\), you can separate the radicand to simplify it into parts. Finally, combine simplified parts to achieve the overall simplified form.