Problem 29 - College Algebra with Modeling and Visualization

Question

Use radical notation to rewrite. $$ a^{-3 / 4} b^{1 / 2} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{b}}{\sqrt[4]{a^3}} \)

Step 1: Rewrite negative exponents using positive exponents

The expression given is \( a^{-3/4} b^{1/2} \). To get rid of the negative exponent for \(a^{-3/4}\), rewrite it as \( \frac{1}{a^{3/4}} \). The expression then becomes \( \frac{b^{1/2}}{a^{3/4}} \).

Step 2: Convert rational exponents to radical form

Recall that a rational exponent \( \frac{n}{m} \) can be written as a radical as \( \sqrt[m]{x^n} \). Thus, converting each factor, we have: \(b^{1/2}\) is \( \sqrt{b} \) and \(a^{3/4}\) is \( \sqrt[4]{a^3} \).

Step 3: Combine the results using radical notation

Substitute the radical forms for their respective bases: \( \frac{b^{1/2}}{a^{3/4}} = \frac{\sqrt{b}}{\sqrt[4]{a^3}} \). This is the expression rewritten in radical notation.

Key Concepts

Negative Exponents

When you see a negative exponent, it might initially seem a bit scary, but it's actually quite simple! A negative exponent means we take the reciprocal of the base. For example, if you have a term like \( a^{-3} \), you would rewrite it as \( \frac{1}{a^3} \). This way, we bring the negative exponent to a positive one by flipping the fraction.

Applying this principle to our exercise, the term \( a^{-3/4} \) becomes \( \frac{1}{a^{3/4}} \). Now, the exponent is positive, making it easier to work with. Remember, negative exponents don't imply a negative number; they simply tell us to take the reciprocal.

Rational Exponents

Rational exponents are expressions where the exponent is a fraction. This might initially appear complex, but understanding them is essential for work in algebra and calculus. A rational exponent like \( a^{m/n} \) can be interpreted as a root and power: it means \( \sqrt[n]{a^m} \). Here, \( n \) is the root you are taking, and \( m \) is the power you are raising things to after finding that root.

In our example, the exponent for \( b \) is \( \frac{1}{2} \), so it reflects a square root, changing \( b^{1/2} \) into \( \sqrt{b} \). For \( a^{3/4} \), this becomes \( \sqrt[4]{a^3} \), representing the fourth root of \( a \) raised to the power of three.

Radical Expressions

Radical expressions involve roots, like square roots, cube roots, and beyond. They are a useful way to express numbers and variables raised to rational exponents in an alternate form.

To transform rational exponents into radical notation, we follow a simple method: exponentiation is replaced with roots. For instance, \( a^{1/2} \) can be written as \( \sqrt{a} \). Similarly, \( x^{m/n} \) is equivalent to \( \sqrt[n]{x^m} \).

When we converted \( b^{1/2} \) to \( \sqrt{b} \) and \( a^{3/4} \) to \( \sqrt[4]{a^3} \), we used radical notation, which often offers a simpler look at the problem. This clearer form helps especially in solving equations, simplifying expressions, or communicating solutions.