Problem 32 - College Algebra with Modeling and Visualization

Question

Use radical notation to rewrite. $$ \left(a^{3 / 4}-b^{3 / 2}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The expression is \( \sqrt[3]{\sqrt[4]{a^3} - \sqrt{b^3}} \).

Step 1: Convert Each Term to Radical Notation

In mathematical terms, a fractional exponent \( \frac{m}{n} \) is equivalent to the \( n \)-th root of the base raised to the \( m \)-th power. Using this rule, we can express \( a^{3/4} \) as \( \sqrt[4]{a^3} \), and \( b^{3/2} \) as \( \sqrt[2]{b^3} \).

Step 2: Apply Fractional Exponent to Entire Expression

Next, address the \((1/3)\) exponent affecting the entire expression. This means we need to take the cube root of the quantity inside the parentheses. Therefore, the expression \((a^{3/4} - b^{3/2})^{1/3}\) becomes \( \sqrt[3]{\sqrt[4]{a^3} - \sqrt{b^3}} \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express roots in algebraic expressions. When you see an exponent written as a fraction, you are actually dealing with roots and powers. For instance, the expression \( a^{3/4} \) translates to the fourth root of \( a \) raised to the third power. Here's how we can understand this:

• The denominator (in this case, 4) tells us the type of root to take, which is the fourth root of \( a \).
• The numerator (3) indicates that we then raise our result to the third power.

So, \( a^{3/4} \) can be expressed as \( \sqrt[4]{a^3} \). Similarly, \( b^{3/2} \) becomes \( \sqrt{b^3} \), because we take the square root (denoted by 2) of \( b \) and raise it to the power of 3.
Using fractional exponents can make simplifying expressions easier and allow for more straightforward manipulation of algebraic terms.

Exponentiation Rules

Exponentiation involves applying certain rules to simplify expressions and equations. Some of the key exponentiation rules include:

• Product of Powers Rule: When multiplying like bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Quotient of Powers Rule: When dividing like bases, subtract their exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent Rule: Any base with an exponent of zero equals one: \( a^0 = 1 \).

Understanding these rules is essential to manipulating expressions efficiently and accurately. In the original exercise, the fractional exponents and radical notation require familiarity with these rules for conversion between different forms. These rules help you navigate through complex expressions and ensure each step is mathematically sound.

Roots in Algebra

Roots in algebra are important for simplifying complex expressions. They are essentially the inverse operation of exponentiation. In simpler terms, a root undoes the operation of raising a number to a power. The most common roots are:

• Square roots, denoted by \( \sqrt{} \), which ask you "what number squared gives this value?"
• Cubic roots, denoted by \( \sqrt[3]{} \), asking "what number cubed gives this value?"
• Higher roots, like fourth roots \( \sqrt[4]{} \), which follow the same logic.

For example, in the exercise, we converted fractional exponents to radical notation. This is because fractional exponents refer to the concept of taking roots:

• a fractional exponent with a denominator of 2 means a square root, 3 means a cube root, and so on.

By converting an expression like \( a^{3/4} \) into \( \sqrt[4]{a^3} \), we understand it as a series of operations involving roots and powers, giving a clear step-by-step way to simplify or evaluate complex expressions. Roots help to translate between different expressions and make complex calculations more manageable.