Problem 19 - Intermediate Algebra

Question

Add or subtract. $$ 2+3 \sqrt{y^{2}}-6 \sqrt{y^{2}}+5 $$

Step-by-Step Solution

Answer

The expression simplifies to \( 7 - 3|y| \).

Step 1: Simplify square roots

Recognize that \( \sqrt{y^2} = |y| \). Therefore, in the expression \( 3\sqrt{y^2} \) and \( 6\sqrt{y^2} \), simplify to \( 3|y| \) and \( 6|y| \), respectively.

Step 2: Substitute simplified square roots back into the expression

Substitute the simplified terms back into the expression, which becomes: \( 2 + 3|y| - 6|y| + 5 \).

Step 3: Combine like terms

Identify and combine like terms: \( 2 + 5 \) into \( 7 \), and \( 3|y| - 6|y| \) into \( -3|y| \). This gives you the expression \( 7 - 3|y| \).

Key Concepts

Understanding Square Roots

Square roots can often seem tricky at first glance, but they're not as confusing as they might appear. A square root of a number is simply a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9.

However, when you're dealing with algebraic expressions, square roots can also involve variables. In the given exercise, we have expressions like \( \sqrt{y^2} \). Here, \( y^2 \) denotes \( y \) multiplied by itself. The square root of \( y^2 \) is \( y \). But since \( y \) could be positive or negative, the proper mathematical expression for this is the absolute value, written as \( |y| \). This accounts for both solutions of the square root, ensuring the result is always non-negative.

The Role of Absolute Value

In mathematics, absolute value plays a crucial role, especially when simplifying expressions involving square roots. Absolute value is essentially a measure of how far a number is from zero, regardless of direction on the number line. Therefore, the absolute value of any number is never negative.

Consider the expression \( \sqrt{y^2} \). As previously discussed, its equivalent is \( |y| \). This tells us that regardless of whether \( y \) is a negative or positive number, the outcome of \( \sqrt{y^2} \) will always be a positive number, or zero. Thus, using absolute value helps ensure that our mathematical expressions remain accurate and simplified.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that makes complex expressions more manageable. "Like terms" are terms that have identical variable parts raised to the same power. In the context of the original exercise, once we've simplified the square roots to express them via absolute values, we have terms like \( 3|y| \) and \( 6|y| \).

To simplify such expressions, you simply add or subtract the coefficients—the numerical parts in front of the terms. For example, in the expression \( 3|y| - 6|y| \), we simply take \( 3 \) minus \( 6 \), resulting in \( -3|y| \). Similarly, you can combine the constant terms \( 2 \) and \( 5 \) to get \( 7 \). By combining like terms, you can tidy up expressions, making them a lot easier to work with and understand.