If y > 0 , which of the following is equal to sqrt(45y^3)?

1. Translate the problem requirements: We need to simplify \(\sqrt{45y^3}\) and express it in the form shown in the answer choices, which all have the pattern: (number)(y to some power)\(\sqrt{\text{number times y to some power}}\)
2. Factor the expression under the radical: Break down both the numerical coefficient (45) and the variable term (\(y^3\)) to identify perfect squares that can be extracted
3. Extract perfect squares from the radical: Remove any perfect square factors from under the radical sign, converting them to their square roots outside the radical
4. Match with answer choices: Compare the simplified form with the given options to identify the correct answer

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we need to do. We have the expression \(\sqrt{45y^3}\) and we need to simplify it into a form that matches one of the answer choices.

Looking at the answer choices, they all follow a similar pattern: they have a number times some power of y, then times the square root of (some number times some power of y).

Our goal is to take \(\sqrt{45y^3}\) and rewrite it in this same format.

Process Skill: TRANSLATE - Converting the radical expression into a format that matches the answer choice patterns

2. Factor the expression under the radical

Now let's break down what's inside the square root: \(45y^3\)

First, let's factor the number 45. What numbers multiply to give us 45?

\(45 = 9 \times 5\)

Notice that 9 is a perfect square (\(9 = 3^2\)), which is great because perfect squares can be "pulled out" of square roots easily.

Next, let's look at \(y^3\). Can we break this down into perfect squares?

\(y^3 = y^2 \times y\)

Here, \(y^2\) is a perfect square, and we have one y left over.

So we can rewrite our expression as:

\(\sqrt{45y^3} = \sqrt{9 \times 5 \times y^2 \times y} = \sqrt{9 \times y^2 \times 5 \times y}\)

3. Extract perfect squares from the radical

Now comes the key step: taking out the perfect squares from under the square root.

Remember that when we have a perfect square under a square root, we can pull it out. For example, \(\sqrt{9} = 3\) and \(\sqrt{y^2} = y\) (since y > 0 as given in the problem).

So from \(\sqrt{9 \times y^2 \times 5 \times y}\), we can pull out:

• \(\sqrt{9}\) becomes 3
• \(\sqrt{y^2}\) becomes y
• What's left under the square root is \(5y\)

Therefore: \(\sqrt{45y^3} = 3 \times y \times \sqrt{5y} = 3y\sqrt{5y}\)

Process Skill: SIMPLIFY - Systematically extracting perfect square factors from the radical

4. Match with answer choices

Our simplified expression is \(3y\sqrt{5y}\).

Looking at the answer choices:

• (A) \(3y\sqrt{5y}\) ✓ This matches exactly!
• (B) \(9y\sqrt{4y}\) ✗ Different numbers
• (C) \(5y\sqrt{3y}\) ✗ Numbers are swapped
• (D) \(3y^2\sqrt{5y}\) ✗ Has \(y^2\) instead of y
• (E) \(5y^2\sqrt{3y}\) ✗ Wrong numbers and has \(y^2\)

We can verify our answer by working backwards: \((3y\sqrt{5y})^2 = (3y)^2 \times (\sqrt{5y})^2 = 9y^2 \times 5y = 45y^3\) ✓

Final Answer

The answer is (A) \(3y\sqrt{5y}\).

We successfully simplified \(\sqrt{45y^3}\) by factoring 45 into \(9 \times 5\), factoring \(y^3\) into \(y^2 \times y\), then extracting the perfect squares (9 and \(y^2\)) from under the radical to get \(3y\sqrt{5y}\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Students may fail to recognize that the goal is to extract perfect squares from under the radical. Instead, they might attempt to use algebraic manipulation or other complex methods, making the problem unnecessarily difficult.

Faltering Point 2: Students may not understand that they need to factor both the numerical coefficient (45) and the variable term (\(y^3\)) separately to identify perfect square components. They might try to work with \(45y^3\) as a single entity without breaking it down.

Errors while executing the approach

Faltering Point 1: When factoring 45, students may incorrectly break it down (for example, as \(15 \times 3\) instead of \(9 \times 5\)) and miss the perfect square factor of 9. This leads to an incorrect simplification.

Faltering Point 2: Students may incorrectly handle the variable term \(y^3\). A common error is writing \(y^3 = y \times y \times y\) and then incorrectly extracting \(\sqrt{y^3} = y\sqrt{y}\) instead of the correct \(\sqrt{y^3} = y\sqrt{y}\), or they may forget that \(y^3 = y^2 \times y\) where \(y^2\) is the perfect square factor.

Faltering Point 3: Students may overlook the given constraint that y > 0. This constraint is crucial because it ensures that \(\sqrt{y^2} = y\) (not \(|y|\)). Without recognizing this, students might hesitate or make errors when extracting y from \(\sqrt{y^2}\).

Errors while selecting the answer

Faltering Point 1: Students may arrive at the correct factorization but then match it incorrectly with the answer choices due to careless reading. For instance, they might confuse \(3y\sqrt{5y}\) with \(5y\sqrt{3y}\) or miss the difference between y and \(y^2\) in the coefficients.