Problem 20 - A Graphical Approach to College Algebra

Question

Answer each question. Which of the following expressions is not simplified? Give the simplified form. A. \(\sqrt[3]{2 y}\) B. \(\frac{\sqrt{5}}{2}\) C. \(\sqrt[4]{m^{3}}\) D. \(\sqrt{\frac{3}{4}}\)

Step-by-Step Solution

Answer

Expression C is not simplified and its simplified form is \(m^{3/4}\).

Step 1: Identify Simplification of Each Expression

To determine which expression is not in a simplified form, we need to check each option one by one. An expression is simplified if there is no further mathematical operation that can be performed to reduce it further.

Step 2: Analyze Expression A

For \(\sqrt[3]{2y}\), check if any further simplification is possible. In cube roots, both the numbers inside, 2 and \(y\), are in their simplest forms. Thus, this expression is already simplified.

Step 3: Analyze Expression B

The expression \(\frac{\sqrt{5}}{2}\) cannot be simplified further because both the numerator and denominator are in their simplest forms and there are no common factors.

Step 4: Analyze Expression C

Consider \(\sqrt[4]{m^{3}}\). Using the property of roots, we can rewrite this as \((m^{3})^{1/4}\), which further simplifies to \(m^{3/4}\). This implies \(\sqrt[4]{m^{3}}\) is not in its simplest form.

Step 5: Analyze Expression D

For \(\sqrt{\frac{3}{4}}\), apply the property that the square root of a fraction is the fraction of the square roots: \(\frac{\sqrt{3}}{\sqrt{4}}\). This becomes \(\frac{\sqrt{3}}{2}\) (since \(\sqrt{4} = 2\)), which is simplified.

Key Concepts

Roots and Radicals

Roots and radicals might seem mysterious at first, but they're all about breaking numbers down into repeat multiplication. When we deal with roots, such as square roots (like \(\sqrt{a}\)) or cube roots (like \(\sqrt[3]{b}\)), we're essentially asking ourselves: 'What number multiplied by itself a certain number of times gives us this original number?'
For instance, the square root of 9, written \(\sqrt{9}\), is 3, because \(3 \times 3 = 9\). Similarly, the cube root of 27, written as \(\sqrt[3]{27}\), is 3, because \(3 \times 3 \times 3 = 27\).
Now, when you see an expression like \(\sqrt[4]{m^{3}}\), it can be tricky. This is a **radical expression** that involves an exponent. We use properties of exponents and roots to transform it: the fourth root is the same as raising it to the power of one-quarter, so \((m^{3})^{1/4}\). This expression needs more simplifying, resulting in \(m^{3/4}\), which is clearer and simpler.
Understanding roots and radicals helps us simplify complex numbers into more graspable pieces. Ordinary numbers, fractions, and even variables can all be broken down using these radical concepts.

• Cubes and roots simplify repeat multiplication.
• They help make complex expressions manageable.
• Applying exponent rules is crucial here.

Fraction Simplification

Fraction simplification is all about making fractions as easy to work with as possible. We want to distill the fraction down to its essence, retaining the same value but in a cleaner form. This often involves reducing the fraction so that the numerator and the denominator have no common factors besides 1.
Take the fraction \(\frac{\sqrt{5}}{2}\) for example. The square root of 5 is an irrational number, meaning it cannot be simplified further into a simple fraction and it does not share any factors with 2, the denominator. This means this expression is already in its simplest form because it's fully reduced.
Another method of fraction simplification involves using the property of radicals. For instance, with expressions like \(\sqrt{\frac{3}{4}}\), we can take the square root of both the numerator and the denominator separately, resulting in \(\frac{\sqrt{3}}{2}\), which is simplification at its best.
Understanding fraction simplification helps make arithmetic and algebra more intuitive, stripping away unnecessary complexities in calculations.

• Simplify by removing common factors.
• Use properties of radicals to ease calculations.

Algebraic Expressions

When diving into algebraic expressions, we're dealing with a combination of numbers, variables, and operations that work together to represent quantities and relationships. These expressions can sometimes seem complex, but breaking them down into smaller parts can reveal their simplicity.
Consider a basic algebraic expression like \(\sqrt[3]{2y}\). Here, we have a cube root applied to a multiplication of a number and a variable. Both elements are in their simplest forms, which tells us this expression is as clean as it gets. Understanding this helps us realize that simplifying isn't just reducing, it's about recognizing when something can't be simplified further.
Another part of simplifying algebraic expressions often involves the use of properties of exponents and roots. The expression \(\sqrt[4]{m^3}\) is a key example of this, where applying the property of roots can transform an expression into a simpler power, \(m^{3/4}\).
Grasping algebraic expressions means getting comfortable with abstraction. It trains our brains to look beyond mere numbers, envisioning possible simplifications and applying operations that reduce complexity.

• Identify when expressions cannot be further simplified.
• Use properties of exponents and roots to your advantage.
• Learn to see simplicity within complexity.