Problem 93 - Algebra A Combined Function

Question

Square where indicated. Simplify if possible. \((5 x)^{2}+(2 y)^{2}\)

Step-by-Step Solution

Answer

\(25x^2 + 4y^2\)

Step 1: Expand the Squared Terms

The given expression is \((5x)^2 + (2y)^2\). Start by expanding each squared term. Use the formula \((a)^2 = a imes a\) to expand each term. This gives:\((5x)^2 = 5x \times 5x = 25x^2\)and\((2y)^2 = 2y \times 2y = 4y^2\).

Step 2: Combine the Expanded Terms

Now, substitute the expanded terms from Step 1 back into the expression. This results in:\(25x^2 + 4y^2\).

Step 3: Simplify the Expression

There are no like terms in the expression \(25x^2 + 4y^2\) to combine, and there is no further simplification possible since each term involves different variables raised to the power of 2.

Key Concepts

Squaring Binomials

Squaring binomials is a fundamental concept in algebra. A binomial is simply a two-term expression, for example,

• \((a + b)\)
• \((5x + 2y)\)

When you "square" a binomial, you multiply it by itself. For example, squaring \((a + b)\) results in \((a + b)(a + b)\). Applying the distributive property leads to:\[a^2 + ab + ab + b^2\]which simplifies to: \[a^2 + 2ab + b^2\]. Notice that when squaring binomials, the result contains:

• The square of the first term (\(a^2\))
• Two times the product of the two terms (\(2ab\))
• The square of the second term (\(b^2\))

Since the original exercise, \((5 x)^2+(2 y)^2\), focused on squaring each term separately, we are not dealing with a typical binomial to be squared fully in one go. However, applying the squaring technique gives the squared terms:

• \((5x)^2 = 25x^2\)
• \((2y)^2 = 4y^2\)

Expanding Expressions

Expanding expressions involves opening up or "spreading out" expressions, especially those that include terms in parentheses raised to powers. To do this correctly, algebraic rules and properties must be used.

When dealing with a single term raised to a power, such as \((5x)^2\), expansion means multiplying the term by itself: \((5x) \times (5x)\), which results in \(25x^2\). Each part of the term—including coefficients and variables—is raised to that power. In this case, both 5 and \(x\) individually get squared, producing \(5^2 imes x^2 = 25x^2\).

The same rule applies for \((2y)^2\), where we expand it as: \(2y \times 2y\), leading to \(4y^2\). Keep in mind that if your initial expression has more than one term in any grouping symbol, you might need to apply more complex methods like distributive property or FOIL (First, Outer, Inner, Last) for binomials.

Combining Like Terms

Combining like terms is an essential step in the simplification of algebraic expressions. It requires the identification and summing of terms with identical variable parts. These are termed "like terms".

For example, \(3x^2 + 2x^2\) can be combined into \(5x^2\), since the variable part, \(x^2\), is the same. The coefficients, or the numerical parts, are simply added.

• Terms such as \(x^2\)and \(y^2\) are not alike.
• They're considered different because of the variable difference.

In the solution \(25x^2 + 4y^2\), the terms cannot be combined because they do not share the same variable component. In combining like terms, always ensure that:

• Only coefficients are combined
• The variable parts remain unchanged

This process helps simplify expressions and is crucial for solving algebraic equations efficiently.