Problem 106 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ -\sqrt{144 t^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \\( -12t^2 \\\).

Step 1: Identify the components of the radical expression

The given expression is \( -\sqrt{144t^4} \). Here, 144 is a perfect square and \( t^4 \) is also a perfect square since its exponent is an even number.

Step 2: Simplify the square root of the number

Calculate the square root of 144. Since 144 is a perfect square, \( \sqrt{144} = 12 \). This simplifies the expression to \( -12\sqrt{t^4} \).

Step 3: Simplify the square root of the variable expression

The expression \( \sqrt{t^4} \) can be rewritten by considering that \( t^4 = (t^2)^2 \). Therefore, the square root of \( t^4 \) is \( t^2 \). Substitute back to get \( -12t^2 \).

Step 4: Complete the simplification

Combine the simplified forms from Step 2 and Step 3 to get the final simplified expression. The expression becomes \( -12t^2 \).

Key Concepts

Perfect Squares

Perfect squares are numbers or expressions that can be expressed as the square of an integer or an algebraic term. For instance, the number 144 is a perfect square because it equals \( 12^2 \). Similarly, an expression like \( t^4 \) is a perfect square since it can be rewritten as \((t^2)^2\). Recognizing perfect squares is crucial in simplifying radical expressions because it allows us to easily find their square roots.When simplifying radical expressions, identifying perfect squares helps to:

• Reduce the complexity of the expression.
• Quickly determine the square root.

Knowing these properties saves time and leads to simplified answers. Practicing identifying which numbers or variables are perfect squares is key to mastering this skill.

Square Root

The square root of a number or an expression is a value that, when multiplied by itself, gives the original number or expression. In our example, the square root of 144 is 12, since \( 12 \times 12 = 144 \). Similarly, the square root of \( t^4 \) is \( t^2 \) because \( (t^2)^2 = t^4 \).Understanding square roots is fundamental in algebra because:

• They simplify the expression under a radical sign.
• They help transition from radical notation to exponent notation.

In algebraic terms, knowing square roots enables further simplification of expressions. Recognizing when and how to compute square roots transforms complex expressions into manageable forms, such as in the expression \( -\sqrt{144t^4} = -12t^2 \). Consistent practice with both numeric and algebraic square roots builds confidence in dealing with various expressions.

Algebraic Expressions

Algebraic expressions can include numbers, variables, and operations such as addition, subtraction, multiplication, and division. They allow us to represent mathematical ideas in a generalized form. For example, \(-\sqrt{144t^4}\) is an algebraic expression containing:

• The number 144, which is a constant.
• The variable \( t \), raised to the 4th power.
• A square root operation, indicated by \( \sqrt{} \).

When simplifying algebraic expressions, especially those under a radical:

• Identify perfect squares and calculate their roots.
• Apply properties of exponents and radicals correctly.

In algebra, simplifying expressions is essential because it makes calculations easier and reduces errors. Practicing on different types of algebraic expressions can lead to skillful manipulation of variables and exponents, essential for solving more complex equations in algebra and calculus.