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Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. In algebra, these expressions allow us to generalize mathematical ideas and solve equations by finding the values of variables. For example, in the expression \(-2 \sqrt{8y^2} + 5 \sqrt{32y^2}\), the terms \(\sqrt{8y^2}\) and \(\sqrt{32y^2}\) are part of an algebraic expression where the variable \(y\) can take on different values.

This particular expression also includes the operation of combining numbers and variables via addition and multiplication. It's important to understand how each piece of the expression relates to each other, which helps in simplifying the expression effectively. Breaking down these parts and understanding their operations can make complex expressions more manageable.

With algebraic expressions, it's crucial to remember:

• The terms are separated by addition or subtraction signs.
• Each term may include constants, coefficients (numbers in front of variables), variables, and operators.

In algebra, one of the key skills is to combine like terms. Like terms are terms in an algebraic expression that have the exact same variable part, meaning both the variable and its exponent must be identical.

For instance, in \(-4 \sqrt{2}y + 20 \sqrt{2}y\), the terms \( -4 \sqrt{2}y \) and \( 20 \sqrt{2}y \) are like terms because they both contain the factor \( \sqrt{2}y \). This makes them combinable by adding or subtracting their coefficients, just like with simple numbers.

To combine these terms:

• Identify the common parts of the terms.
• Add or subtract the coefficients of these like terms.

This results in a simpler expression: \[( -4 + 20 ) \sqrt{2}y = 16 \sqrt{2}y\]

Remember, combining like terms helps to simplify expressions, making equations easier to solve.

Simplifying square roots is the process of reducing them to their simplest form. This usually involves pulling perfect squares out of the square root.

• Find the factors inside the square root that are perfect squares.
• Take the square root of these perfect squares and place them outside the radical sign.

Let's look at \(-2 \sqrt{8y^2}\) and \(+ 5 \sqrt{32y^2}\):

First, for \(\sqrt{8y^2}\):

• Identify that \(8y^2 = (4)(2y^2)\) where 4 is a perfect square.
• This simplifies to \(2 \sqrt{2y^2}\).

Next, simplify \(\sqrt{32y^2}\):

• This can be written as \(32y^2 = (16)(2y^2)\), since 16 is a perfect square.
• This reduces to \(4 \sqrt{2y^2}\).

Thus, after simplification:

• \(-2 \times 2 \sqrt{2y^2}\) becomes \(-4\sqrt{2}y\).
• \(+ 5 \times 4 \sqrt{2y^2}\) becomes \(+ 20\sqrt{2}y\).

Understanding and practicing these steps makes simplifying square roots more intuitive and is a useful skill in dealing with various algebraic problems.