Problem 8 - Contemporary Precalculus

Question

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{\sqrt{a b} \sqrt[3]{a b^{4}}}{\sqrt{a}(\sqrt[3]{b})^{4}}$$

Step-by-Step Solution

Answer

**Question:** Simplify the following expression, given that \(a, b, c,\) and \(d\) are all positive: $$\frac{\sqrt{a b} \sqrt[3]{a b^{4}}}{\sqrt{a}(\sqrt[3]{b})^{4}}$$ **Short answer:** $$(ab)^{\frac{1}{6}}$$

Step 1: Write the roots as exponential expressions

The given expression is: $$\frac{\sqrt{a b} \sqrt[3]{a b^{4}}}{\sqrt{a}(\sqrt[3]{b})^{4}}$$ The square root of a number is the same as raising it to the power of 1/2, and the cube root is the same as raising it to the power of 1/3. Therefore, we can rewrite the initial expression as: $$\frac{{(a b)^{\frac{1}{2}}} {(a b^{4})^{\frac{1}{3}}}}{{(a)^{\frac{1}{2}}}(b^{\frac{4}{3}})})$$

Step 2: Use exponent rules to simplify the expression

First, let's apply the exponent rule for powers of products, which states that \({(ab)^x} = {a^x b^x}\). We can rewrite the expression as: $$\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} a^{\frac{1}{3}} b^{\frac{4}{3}}}{a^{\frac{1}{2}} b^{\frac{4}{3}}}$$ Now, utilize the exponent rule for multiplication: \(a^x a^y = a^{x + y}\) to merge the terms with the same base (\(a\) and \(b\) in this case). We get: $$\frac{a^{\frac{1}{2}+\frac{1}{3}} b^{\frac{1}{2}+\frac{4}{3}}}{a^{\frac{1}{2}} b^{\frac{4}{3}}}$$

Step 3: Simplify the exponents

Combine the exponent fractions for \(a\) and \(b\) and simplify: $$\frac{a^{\frac{5}{6}} b^{\frac{11}{6}}}{a^{\frac{1}{2}} b^{\frac{4}{3}}}$$

Step 4: Use the exponent rule for division to simplify the expression

Use the exponent rule for division, which states that \(\frac{a^x}{a^y} = a^{x - y}\), to simplify the expression: $$a^{\frac{5}{6} - \frac{1}{2}} b^{\frac{11}{6} - \frac{4}{3}}$$ Simplify the exponents: $$a^{\frac{1}{6}} b^{\frac{1}{6}}$$

Step 5: Combine the terms

Since the exponents for \(a\) and \(b\) are the same, we can use the exponent rule for products to combine them: $$(ab)^{\frac{1}{6}}$$ Therefore, the simplified form of the given expression is: $$\boxed{(ab)^{\frac{1}{6}}}$$

Key Concepts

Exponent Rules

When dealing with expressions containing exponents, it's essential to understand the basic exponent rules to simplify them.
Exponent rules help us manipulate expressions involving powers, making them more manageable. Here are some key things to remember:

• Multiplying like bases: When you multiply terms with the same base, you simply add the exponents: \(a^m \times a^n = a^{m+n}\).
• Dividing like bases: When dividing, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Powers of a power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Powers of a product: Distribute the exponent to each base: \((ab)^n = a^n \times b^n\).

Using these rules in combination can help you tackle more complex expressions, making them significantly simpler. As seen in the solution, applying these rules step-by-step helps achieve a single term with the base \((ab)\) raised to a smaller power.

Rational Exponents

Rational exponents can seem confusing at first, but they're simply another way to express roots.
A rational exponent, like \(a^{\frac{m}{n}}\), implies the \(n\)-th root of \(a\) raised to the \(m\)-th power. So, \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Breaking it down:

• The denominator \(n\) of the exponent indicates the type of root (e.g., cube root for 3).
• The numerator \(m\) represents the power the base is raised to once the root is applied.

This notation allows us to work with roots using the same rules as powers.
So for example, \(\sqrt{a} = a^{\frac{1}{2}}\) and \(\sqrt[3]{b} = b^{\frac{1}{3}}\). When simplifying radical expressions or combining them with other terms, changing roots to rational exponents helps apply the exponent rules effectively.

Radicals

Radicals represent root operations in mathematics.
Most commonly, you'll come across square roots (\(\sqrt{}\)) and cube roots (\(\sqrt[3]{}\)), but radicals can extend to any degree. When we want to simplify expressions, converting these radicals to rational exponents allows more straightforward calculations using exponent rules.
Here’s how it typically works:

• Square root: Denoted as \(\sqrt{a}\), equals \(a^{\frac{1}{2}}\).
• Cube root: Denoted as \(\sqrt[3]{c}\), equals \(c^{\frac{1}{3}}\).

Once in rational exponent form, apply the exponent rules to simplify further.
For complex expressions like our given exercise, the transformation into rational exponents helps break down the expression step-by-step until you reach a simplified form, as shown in the solution. Mastering radicals and rational exponents means you can navigate between forms fluidly, ensuring you're ready to simplify or solve any related problem.