Problem 61 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (3 y-\sqrt{7 y})^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $9y^2 + 7y - 6y\sqrt{7y}$.

Step 1: Apply the Squaring Operation

The first step involves squaring the given parentheses expression. This is equivalent to multiplying the expression by itself. So, we will multiply \((3y - \sqrt{7y})(3y - \sqrt{7y})\).

Step 2: Expand the Multiplication

To expand the multiplication, we will apply the distributive property and multiply each term in the first expression by each term in the second expression. So, we have: $$(3y - \sqrt{7y})(3y - \sqrt{7y}) = (3y)(3y) + (3y)(-\sqrt{7y}) + (-\sqrt{7y})(3y) + (-\sqrt{7y})(-\sqrt{7y})$$

Step 3: Simplify the Terms

Next, we will simplify the terms of the expanded expression: $$(3y)(3y) = 9y^2$$ $$(3y)(-\sqrt{7y}) = -3y\sqrt{7y}$$ $$(-\sqrt{7y})(3y) = -3y\sqrt{7y}$$ $$(-\sqrt{7y})(-\sqrt{7y}) = 7y$$

Step 4: Combine the Simplified Terms

Now, let's combine the simplified terms: $$9y^2 - 3y\sqrt{7y} - 3y\sqrt{7y} + 7y$$

Step 5: Simplify the Expression

Lastly, we will simplify the combined expression by combining like terms: $$9y^2 + 7y - 6y\sqrt{7y}$$ The simplified expression is: $$ (3y - \sqrt{7y})^2 = 9y^2 + 7y - 6y\sqrt{7y} $$

Key Concepts

Distributive Property

Understanding the distributive property is fundamental when dealing with algebraic expressions, especially those involving binomials. The distributive property allows us to multiply a single term across terms inside parentheses.
For example, in the expression \(a(b+c)\), the term \(a\) is distributed to both \(b\) and \(c\). This gives us \(ab + ac\).
When squaring a binomial such as \(3y - \sqrt{7y}\), you're essentially multiplying the binomial by itself. Using the distributive property helps in systematically carrying out each multiplication involved in this process:

• Apply the property to distribute the first term of the first expression to both terms of the second expression.
• Next, distribute the second term of the first expression to both terms of the second expression.

This results in the equation:
\((3y)(3y) + (3y)(-\sqrt{7y}) + (-\sqrt{7y})(3y) + (-\sqrt{7y})(-\sqrt{7y})\)
Mastering this property ensures you can expand and simplify polynomial expressions accurately.

Radical Expressions

Radical expressions involve roots, and in this context, we have a square root, often denoted with the symbol \(\sqrt{}\). They can sometimes make algebraic manipulations seem complex, but understanding how to work with them is crucial.
When you multiply radical expressions, such as when we got \(-\sqrt{7y})(3y) \) or \((-\sqrt{7y})(-\sqrt{7y})\), you apply the same rules as you do with simple numbers. Let's break it down further:

• For the product \((-\sqrt{7y})(3y)\), you multiply the coefficients and keep the radical part intact, giving you \(-3y\sqrt{7y}\).
• When the same radical is multiplied by itself, such as \((-\sqrt{7y})(-\sqrt{7y})\), the radicals cancel out, resulting in the expression under the radical: \(7y\).

This manipulation of radicals is essential for simplifying and solving equations involving roots.

Polynomial Simplification

Simplifying polynomials involves combining like terms and eliminating redundant parts to produce a more straightforward expression. Once you've expanded a polynomial using the distributive property, as we did in the previous steps, simplification is the final and essential task.
Let's recap the product we expanded: \((3y - \sqrt{7y})^2\) became \((9y^2 - 3y\sqrt{7y} - 3y\sqrt{7y} + 7y)\).

The simplification process includes:

• Identifying and combining like terms. In our expansion, \(-3y\sqrt{7y}\) and \(-3y\sqrt{7y}\) are like terms, which combine to \(-6y\sqrt{7y}\).
• Rewriting the expression in a neatly ordered form to improve clarity. This generally means arranging terms from the highest power downwards.

The simplified result gives us: \(9y^2 + 7y - 6y\sqrt{7y}\).
Being adept at polynomial simplification helps streamline expressions, making equations easier to solve and understand.