Problem 19
Question
Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 2+3 \sqrt{y^{2}}-6 \sqrt{y^{2}}+5 $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( 7 - 3y \).
1Step 1: Identifying like terms
First, identify the like terms in the given expression. We have constants (2 and 5) and square root terms with the same expression inside the radical, denoted as \( \sqrt{y^2} \). The terms under the square root, \( \sqrt{y^2} \), simplify to \( y \) since \( y \) is a positive real number.
2Step 2: Simplifying square root terms
Simplify the square root terms by substituting \( \sqrt{y^2} \) with \( y \). Thus, we can rewrite the expression as \(2 + 3y - 6y + 5 \).
3Step 3: Combining like terms
Combine the like terms. First, address the constants: \( 2 + 5 = 7 \). Next, address the \( y \) terms: \(3y - 6y = -3y\).
4Step 4: Writing the final expression
Combine the results from the previous step to express the simplified form of the original expression. This results in \( 7 - 3y \).
Key Concepts
Simplifying ExpressionsSquare RootsCombining Like Terms
Simplifying Expressions
Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form. This makes them easier to work with and understand. When simplifying any algebraic expression, whether it involves variables, numbers, or both, the goal is to perform operations that remove any unnecessary complexity.
For example, any operation that combines constants or variables should be completed. Adding and subtracting numbers, collecting like terms, and simplifying radicals are all part of this process. In our exercise, we simplified the square root term \( \sqrt{y^2}\), revealing it as \( y \), and then processed other parts of the expression to achieve a neat, consolidated final result.
The process helps in tasks such as solving equations, graphing functions, and analyzing mathematical situations. Mastery of simplification makes understanding more complex algebraic operations easier and is a foundational skill in mathematics.
For example, any operation that combines constants or variables should be completed. Adding and subtracting numbers, collecting like terms, and simplifying radicals are all part of this process. In our exercise, we simplified the square root term \( \sqrt{y^2}\), revealing it as \( y \), and then processed other parts of the expression to achieve a neat, consolidated final result.
The process helps in tasks such as solving equations, graphing functions, and analyzing mathematical situations. Mastery of simplification makes understanding more complex algebraic operations easier and is a foundational skill in mathematics.
Square Roots
Square roots often appear in algebra, particularly in simplifying expressions. A square root essentially asks, "What number multiplied by itself gives us this number?" For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).
In the given exercise, we simplified \( \sqrt{y^2} \). The square inside the root sign and the square root itself effectively cancel each other out, leaving us with \( y \). This is because \( y \), when squared and then rooted, returns to its original form.
When working with expressions that contain square roots, focusing on the value inside the radical (the radicand) is crucial. Understanding how to manipulate and simplify these components is key to mastering problems involving roots. Remember, dealing with square roots requires them to be simplified whenever possible, turning complex expressions into simpler, more manageable ones.
In the given exercise, we simplified \( \sqrt{y^2} \). The square inside the root sign and the square root itself effectively cancel each other out, leaving us with \( y \). This is because \( y \), when squared and then rooted, returns to its original form.
When working with expressions that contain square roots, focusing on the value inside the radical (the radicand) is crucial. Understanding how to manipulate and simplify these components is key to mastering problems involving roots. Remember, dealing with square roots requires them to be simplified whenever possible, turning complex expressions into simpler, more manageable ones.
Combining Like Terms
Combining like terms is an essential technique in algebra used to simplify expressions by merging terms with the same variables or constants. When we say terms are 'like,' we mean they share the same variable components, even if they're multiplied by different numbers (coefficients).
In our exercise, you have terms like \( 3y \) and \( -6y \). These are like terms because they both have the variable \( y \). We combine them by adding their coefficients: \( 3 - 6 \), which results in \( -3 \).
Similarly, numbers without variables, known as constants, are also combined. In our instance, \( 2 + 5 \) gives us \( 7 \). This straightforward process vastly simplifies expressions, making them easier to work with. Mastering this concept reduces errors and helps in effectively solving equations.
In our exercise, you have terms like \( 3y \) and \( -6y \). These are like terms because they both have the variable \( y \). We combine them by adding their coefficients: \( 3 - 6 \), which results in \( -3 \).
Similarly, numbers without variables, known as constants, are also combined. In our instance, \( 2 + 5 \) gives us \( 7 \). This straightforward process vastly simplifies expressions, making them easier to work with. Mastering this concept reduces errors and helps in effectively solving equations.
Other exercises in this chapter
Problem 19
Use a calculator to approximate each square root to three decimal places. Check to see that each approximation is reasonable. $$ \sqrt{7} $$
View solution Problem 19
Use radical notation to rewrite each expression. Simplify if possible. $$ (-64)^{2 / 3} $$
View solution Problem 19
Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{4}{27}} $$
View solution Problem 19
Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (4-7 i)+(2+3 i) $$
View solution