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Chapter 0: Problem 59

Simplify each expression. \(\sqrt{y^{2}}\)

Short Answer

Expert verified
y

Step by step solution

01

Understand the Problem

The task is to simplify the expression \( \sqrt{y^{2}} \). Recognize that this expression involves the square root of a square.
02

Recall the Definition of a Square Root

\( \sqrt{y^{2}} \) represents a value that, when squared, gives \( y^{2} \).
03

Simplify the Expression

A square root undoes a square. Therefore, \( \sqrt{y^{2}} = y \). Since squaring a number and then taking the square root brings you back to the original number (assuming the number is non-negative), \ y \ is the simplified form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols such as +, -, *, and /.
Variables are symbols used to represent unknown values and can be letters like x, y, or z.
For example, in the expression \( \sqrt{y^{2}} \), y is the variable. Algebraic expressions can be simple, like a single variable or number, or more complex, involving multiple variables and operations.
It's crucial to understand how to manipulate these expressions correctly, using rules and properties of algebra.
Square Roots
Square roots are mathematical operations that find a number which, when multiplied by itself, gives the original number.
For example, the square root of 9 is 3 because \(3 \cdot 3 = 9 \).
In algebra, the square root of a variable squared simplifies to the variable itself, provided the variable represents a non-negative value.
This concept is used frequently to simplify expressions like \( \sqrt{y^{2}} \).
In this case, because \( \sqrt{y^{2}} \) represents a value that squared gives \( y^{2} \,\ \), it simplifies directly to y.
Simplification
Simplification in algebra involves reducing an expression to its most basic form. It often means combining like terms, using arithmetic operations, or applying algebraic rules like the properties of exponents and roots.
For the expression \( \sqrt{y^{2}} \), simplification involves recognizing that the square root and the square cancel each other out.
This results in the expression simplifying to just y.
Simplifying expressions makes them easier to work with and understand, and is a fundamental skill in algebra.

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Most popular questions from this chapter

Multiply and simplify. $$ 8 p^{5}\left(-2 p^{2}-5 p-1\right) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 5 \sqrt{7}(3 \sqrt{7}-2 \sqrt{5}+3) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{a+2}-\sqrt{a-2})^{2} $$

Perform the indicated operations and simplify. $$ (5 m-3)^{2} $$

Simplify each expression. $$ (x-4)^{2}-6(x+1)^{2} $$
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