Problem 25 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt[4]{5 a^{3}} \sqrt[4]{125 a^{2}} $$

Step-by-Step Solution

Answer

5a\sqrt[4]{a}.

Step 1: Multiply Inside the Radicals

When multiplying radicals of the same root, we can multiply the numbers inside the radicals together. Thus, combining the terms inside each radical:\[ \sqrt[4]{5a^3} \cdot \sqrt[4]{125a^2} = \sqrt[4]{(5a^3) \times (125a^2)}\]First, multiply the numbers: \[5 \times 125 = 625.\]Then, multiply the powers of \(a\): \[a^3 \times a^2 = a^{3+2} = a^5.\]

Step 2: Take the Fourth Root

Now apply the fourth root to the product we obtained:\[ \sqrt[4]{625a^5}.\]We need to simplify this expression. Start by finding the fourth root of 625. Since \(625 = 5^4\), the fourth root of \(625\) is 5.

Step 3: Simplify the Variable Part with Fourth Root

Now take the fourth root of the variable part\[ a^5. \]We can rewrite the expression \(a^5\) under the fourth root as:\[ \sqrt[4]{a^4 \cdot a^1}.\]The fourth root of \(a^4\) is \(a\), leaving us with \(a\) multiplied by the fourth root of \(a\):\[ a \cdot \sqrt[4]{a}.\]

Step 4: Combine All Components

Combine the fourth roots simplified earlier:\[ 5a \cdot \sqrt[4]{a}.\]That's our final expression after multiplying and simplifying the given radicals.

Key Concepts

Fourth Root

The fourth root of a number is what you multiply by itself four times to get the original number. For example, the fourth root of 16 is 2, because multiplying 2 by itself four times (2 x 2 x 2 x 2) equals 16. In mathematical terms, this is represented as \( x^{1/4} \).

To find the fourth root, you can think of it as searching for the number that, when raised to the power of four, gives you the original value. This is especially useful when dealing with higher exponents or large numbers. When taking the fourth root of a variable, you distribute the root over each factor separately, ensuring simpler expressions.

For instance, if you have \( \ a^5 \), taking the fourth root involves separating it into \( a^4 \times a^1 \). The fourth root of \( a^4 \) is \( a \), leaving you with \( a \times \sqrt[4]{a} \) for the expression.

Simplifying Radicals

Simplifying radicals involves reducing the expressions under the radical sign to their simplest form. When you simplify radicals, you try to get the lowest possible numerical value or reduce the variable expression as much as possible.

The simplification process often involves factoring the number under the radical into its prime factors when dealing with numbers, and using exponent rules for variables. If factorable into a perfect power (e.g., square, cube, or fourth), those are removed from under the radical. For example, the number 625 can be expressed as \( 5^4 \). Therefore, \( \sqrt[4]{625} \) simplifies to 5.

When simplifying variables, like \( a^5 \) under a fourth root, you look to express them as a perfect power and a leftover, such as \( \sqrt[4]{a^4 \times a^1} \). This becomes \( a \times \sqrt[4]{a} \), since \( a^4 \) simplifies to \( a \) under the fourth root.

Multiplying Radicals

Multiplying radicals involves multiplying the numbers and variables under the radical signs and then simplifying the result. It's crucial to remember that you can directly multiply under the radicals when their indices are the same.

For example, when multiplying \( \sqrt[4]{5a^3} \cdot \sqrt[4]{125a^2} \), you pair up the numbers and the variables, giving you \( \sqrt[4]{(5 \cdot 125) \cdot (a^3 \cdot a^2)} \). Once under a single radical sign, you multiply the numbers and variables, resulting in \( \sqrt[4]{625a^5} \).

This combined radical can then be simplified further. The main point is ensuring that you consolidate under one radical and then simplify by separating values that can be expressed as perfect fourth powers, as seen with 625, and breaking down the variables into portions that simplify smoothly.