Problem 26 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt[4]{2 r^{3}} \sqrt[4]{8 r^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2r^{5/4}\).

Step 1: Express the Radicals with Exponents

Start by rewriting the radicals using fractional exponents. The expression becomes: \[ (2r^3)^{1/4} imes (8r^2)^{1/4} \] as each 4th root can be represented as raising to the power of 1/4.

Step 2: Combine the Bases

Combine the bases under a single radical as they are raised to the same power. This gives: \[ ((2r^3) imes (8r^2))^{1/4} \]

Step 3: Simplify Inside the Radical

Multiply the numbers and the variables together inside the radical: \[ = (16r^{5})^{1/4} \] because 2 times 8 is 16, and when you multiply \(r^3\) and \(r^2\), you get \(r^{5}\).

Step 4: Apply the Exponent Rule

Distribute the power \(1/4\) to both the number and the variable inside the radical: \[ = 16^{1/4} imes (r^{5})^{1/4} \]

Step 5: Simplify Each Part

Calculate each part separately:- \(16^{1/4} = 2\) because the 4th root of 16 is 2.- \(r^{5/4}\) because when you multiply exponents, you keep the base and multiply exponents: \((r^{5})^{1/4} = r^{5/4}\).

Step 6: Combine and Simplify the Expression

Now, combine the simplified parts:\[ = 2 imes r^{5/4} \] Thus, the simplified form of the original expression is \(2r^{5/4}\).

Key Concepts

Simplifying Radicals

When it comes to radicals, simplifying them involves rewriting the expression in the most concise and manageable form. Radicals are often associated with roots, such as square roots or fourth roots. In our given exercise, we had two fourth roots that we needed to simplify. Initially, this may appear challenging, but by using a systematic approach, it becomes straightforward.

To simplify radicals, a good starting point is to express the radical expressions in terms of fractional exponents. For example, the fourth root of any number can be transformed into an exponent of 1/4. This makes it easier to apply algebraic rules as seen in our exercise where we started with radicals such as \(\sqrt[4]{2r^{3}}\) and expressed them in a simpler form.

• Rewrite radicals using fractional exponents, for instance, the fourth root can be written as a power of \(1/4\).
• Combine like bases if they are raised to the same power. This step simplifies the multiplication of expressions under the same radical.
• If possible, separate numbers and variables to make calculations easier.

By following these steps, simplifying radicals becomes less intimidating and much more manageable.

Fractional Exponents

Fractional exponents provide us with a handy way of dealing with roots and radicals in an expression. These exponents help us to rewrite root expressions using powers, which can simplify calculations greatly. In the context of the original exercise, the expression \(\sqrt[4]{2r^{3}}\) was rewritten as \((2r^{3})^{1/4}\). This transformation is crucial for simplifying and multiplying radicals.

Fractional exponents have a straightforward meaning: the denominator represents the root, and the numerator represents the power of the number. Therefore, an exponent such as \(x^{a/b}\) implies taking the b-th root of \(x\) and then raising to the power of \(a\).

• Identify the root and power from the fractional exponent: numerator for power, denominator for root.
• Apply exponent multiplication principles: when raising a power to another power, you multiply the exponents.

This method was useful in our exercise to combine and simplify parts, resulting comfortably in expressions easier to handle and solve.

Multiplying Radicals

Multiplying radicals often takes advantage of the same properties of exponents and bases. This process involves multiplying the numbers inside the radical signs when they have like bases and converting these operations into simpler form step-by-step.

In our exercise, the task was to multiply two fourth-root expressions: \(\sqrt[4]{2r^{3}}\) and \(\sqrt[4]{8r^{2}}\). We formulated these into an exponent-based format before proceeding to multiply. Using the property that allows the multiplication of coefficients and like bases under the same radical simplifies the result, transforming our expression into \((16r^{5})^{1/4}\).

• Convert radical expressions to fractional exponents to leverage multiplication properties.
• Combine coefficients and variables separately; multiply like bases, combine exponents, and simplify.
• Return the results to a suitable form, such as a simplified radical or expression.

Adopting these techniques ensures accurate and simplified outcomes, easing the process of managing complex radical multiplications.