Problem 11 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ \left(9 x^{4}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

\(3x^2\)

Step 1: Understanding the Expression

The given expression is \( \left(9 x^{4}\right)^{1/2} \). Here, \( 1/2 \) denotes the square root. So, we need to express this using radical notation.

Step 2: Applying Radical Notation

The expression \( \left(a\right)^{1/2} \) can be rewritten as \( \sqrt{a} \). Applying this to \( \left(9 x^{4}\right)^{1/2} \), we rewrite it as \( \sqrt{9x^4} \).

Step 3: Simplifying the Radical Expression

Now, we need to simplify \( \sqrt{9x^4} \). Split it into two separate square roots: \( \sqrt{9} \) and \( \sqrt{x^4} \).

Step 4: Simplifying \( \sqrt{9} \)

The square root of 9 is 3 because \( 3^2 = 9 \). Thus, \( \sqrt{9} = 3 \).

Step 5: Simplifying \( \sqrt{x^4} \)

The square root of \( x^4 \) is \( x^2 \), because \( (x^2)^2 = x^4 \). So, \( \sqrt{x^4} = x^2 \).

Step 6: Combining the Simplified Terms

Combine the simplified terms together: \( \sqrt{9x^4} = 3x^2 \). This is the simplified version of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions means to make them easier to understand or work with by reducing their complexity. In algebra, this often involves combining like terms or reducing expressions to their simplest form. Consider the example expression \( \sqrt{9x^4} \). Here, we split the expression into parts, like \( \sqrt{9} \) and \( \sqrt{x^4} \), and simplify each part individually.

• Breaking down complex parts: Identify and simplify square roots or other operations.
• Combining like terms: Gather similar elements to make calculations easier.
• Resulting expression: Simplified final form is more manageable, like \( 3x^2 \).

Simplifying transforms a challenging problem into an accessible one, making further operations or calculations easier.

Square Roots

A square root refers to a value that, when multiplied by itself, gives the original number. Consider \( \sqrt{9} \); it equals 3 because \( 3^2 = 9 \). Recognizing square roots is crucial in algebra, especially in expressions involving exponents.

• Simple square roots: Numbers like \( 4 \) (where \( \sqrt{4} = 2 \)) and \( 16 \) (where \( \sqrt{16} = 4 \)).
• Exponential roots: For \( x^4 \), \( \sqrt{x^4} = x^2 \) since \((x^2)^2 = x^4 \).
• Managing variables: Apply square root principles to expressions with variables for simplification.

Square roots help in condensing expressions and are vital for radical simplification.

Exponents

Exponents indicate how many times a number (base) is multiplied by itself. For example, in \( x^4 \), 4 is the exponent telling us that \( x \) is multiplied by itself four times.

• Power operations: Simplifying involves understanding power rules, like \((x^m)^n = x^{m\times n}\).
• Root interpretation: The square root entails an exponent of 1/2, transforming expressions into manageable forms.
• Simplification: For \( x^4 \), \( \sqrt{x^4} \) simplifies using exponents to \( x^2 \).

Grasping exponents is essential to handle expressions with powers and to simplify radical expressions effectively.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations, such as addition or multiplication, to create expressions like \( 9x^4 \). They allow us to represent mathematical ideas symbolically.

• Components: Composed of terms like coefficients (e.g., 9) and variables with exponents (e.g., \( x^4 \)).
• Combining and simplifying: Utilize operations and identities for merging like terms and reducing complexity.
• Practical application: Used in equations to solve for unknown variables.

Algebraic expressions form the backbone of algebra, promoting flexible problem-solving techniques and enabling mathematical reasoning.