Problem 88 - Algebra 2

Question

Which expression is NOT equivalent to \(\sqrt[4]{4 n^{2}} ?\) \(\begin{array}{llll}{\text { A. }\left(4 n^{2}\right)^{\frac{1}{4}}} & {\text { B. } 2 n^{\frac{1}{2}}} & {\text { C. }(2 n)^{\frac{1}{2}}} & {\text { D. } \sqrt{2 n}}\end{array}\)

Step-by-Step Solution

Answer

Options C and D are NOT equivalent to \(\sqrt[4]{4 n^{2}}\). Therefore, the expressions presented in options C and D are not equivalent to the initial given expression.

Step 1: Rewrite the main expression

First, rewrite the main expression, \(\sqrt[4]{4 n^{2}}\), as a power to better compare with the given options. This can be written as \((4 n^{2})^{\frac{1}{4}}\).

Step 2: Break down the main expression

According to the laws of exponents, \((a b)^n = a^n b^n\). Applying this rule to our expression gives us \((4^{\frac{1}{4}}) * (n^{2})^{\frac{1}{4}}\). This simplifies to \(2 * n^{\frac{1}{2}}\) because the fourth root of 4 is 2 and \(n^{2 * \frac{1}{4}} = n^{\frac{1}{2}}\). Also, rearrange the expression to \(2n ^{\frac{1}{2}}\). This is equivalent to option B.

Step 3: Compare with other options

The other options C and D do not match the simplified expression obtained. Option C equals \(2*\sqrt{n} = 2*n^{\frac{1}{2}}\) which is not equivalent to our main expression \(2n^{\frac{1}{2}}\). For option D, we have \(\sqrt{2n}\), which is equivalent to \(2n^{\frac{1}{2}}\) which is also not equivalent to our main expression. Therefore, options C and D do not match our simplified expression and are not equivalent to the initial given expression \(\sqrt[4]{4 n^{2}}\).

Key Concepts

Understanding Exponents

Exponents are a mathematical operation that involves raising numbers to a power. When we say a number like \( n^2 \), it means \( n \) is multiplied by itself once, making \( n \times n \). The number above the base, called the exponent, tells us how many times to multiply the base. This can greatly simplify repeated multiplication.
For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \), resulting in 81. Exponents are crucial in algebra for expressing large numbers concisely.

• **Base**: The number being multiplied.
• **Exponent**: Shows how many times the base is used in multiplication.
• **Power**: The entire expression (e.g., \( 3^4 \)).

Understanding this concept makes it easier to work with complex equations and understand how equivalent expressions come into play in algebra.

Exploring Equivalent Expressions

An equivalent expression in algebra means that two expressions, though looking different, hold the same value when calculated. Finding equivalent expressions is fundamental for solving algebraic equations and simplifying problems.
To determine if two expressions are equivalent, simplify each one and compare. If both expressions reduce to the same simple form, they are equivalent. Consider our exercise, where we are tasked to find which expression is not equivalent to \(\sqrt[4]{4 n^2}\). This can be rewritten in different ways using exponents, yet may or may not be equivalent depending on manipulation.

• **Simplification**: Reducing expressions to simpler terms.
• **Comparison**: Breaking down and comparing simplified forms.
  

Finding equivalent expressions helps in identifying the correct forms in algebraic equations, simplifying and comparing them for better understanding.

Laws of Exponents

The laws of exponents are rules that govern how we perform operations involving exponents. These laws make computation manageable and allow us to transform expressions.
Here are some fundamental laws of exponents:

• **Product of Powers Rule**: \( a^m \times a^n = a^{m+n} \)
• **Power of a Power Rule**: \( (a^m)^n = a^{m \times n} \)
• **Power of a Product Rule**: \( (ab)^n = a^n \times b^n \)
• **Quotient of Powers Rule**: \( \frac{a^m}{a^n} = a^{m-n} \)

Applying these principles to our problem: we've been asked to identify which expression isn’t equivalent to \(\sqrt[4]{4 n^2}\). Using these rules, we see \((4n^2)^{\frac{1}{4}}\) becomes a product of powers, simplifying the process, and helping identify incorrectly equivalent options from the set of given expressions.