Problem 25 - Elementary and 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

The product is \( 5 a^{5/4} \).

Step 1: Simplify Each Radical Expression

Start by simplifying each radical separately. For the first radical, \( \sqrt[4]{5 a^3} \), express it as \( (5 a^3)^{1/4} \). For the second radical, \( \sqrt[4]{125 a^2} \), express it as \( (125 a^2)^{1/4} \).

Step 2: Break Down the Radicals

Rewrite each radical using properties of exponents: \( (5 a^3)^{1/4} = 5^{1/4} a^{3/4} \) and \( (125 a^2)^{1/4} = 125^{1/4} a^{2/4} \). Now, simplify: \(125 = 5^3\), so \( 125^{1/4} = (5^3)^{1/4} = 5^{3/4} \). This gives \( 125^{1/4} = 5^{3/4} \).

Step 3: Multiply the Simplified Expressions

Multiply the simplified expressions: \( 5^{1/4} a^{3/4} \times 5^{3/4} a^{2/4} = 5^{(1/4 + 3/4)} a^{(3/4 + 2/4)} \).

Step 4: Simplify the Final Expression

Add the exponents of like bases: \( 5^{1} a^{5/4} \). Since \( 5^{1} = 5 \), the simplified expression is \( 5 a^{5/4} \).

Key Concepts

Multiplication of radicals

When encountering radical expressions, multiplying them involves a few simple steps that make life easier. First, let's recognize each radical's form. In our example, we have

• \( \sqrt[4]{5 a^3} \)
• \( \sqrt[4]{125 a^2} \)

The multiplication of radicals can be simplified by expressing each under a single radical if they share the same index. In this case, both radicals are fourth roots, allowing us to combine them into\( \sqrt[4]{5 a^3 \times 125 a^2} \). Breaking it down further, the expression becomes one single larger expression under the radical.

Properties of exponents

Exponents have key properties that simplify the manipulation of algebraic expressions. These properties are essential when dealing with radicals. The most crucial rules to remember when dealing with the multiplication of exponents include:

• When multiplying like bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

In the given expression, recognize that each term like \( (5 a^3)^{1/4} \) can be separated into individual factors of 5 and \( a \). Applying the properties, the 125 can be rewritten as \( 5^3 \), enabling the use of these rules to combine and rationalize the expression further.

Algebraic expressions

Handling algebraic expressions requires substituting radicals for exponents since they offer a clearer path to simplification. When you have terms like \( \sqrt[4]{5 a^3} \), represent them with fractional exponents since it's equivalent to \( (5 a^3)^{1/4} \). This representation is particularly helpful in making multiplication straightforward.
Algebraic expressions often involve variables, which in this problem are handled as well by assuming they are all positive real numbers. This guarantees that operations such as simplification and combination can proceed without worrying about undefined expressions or negative roots. By simplifying step by step, keep the expression in its simplest form when merging like terms or when adding exponents.

Fourth roots

The concept of a fourth root might seem complex, but it is essentially asking what number needs to be multiplied four times to achieve the original number. In notation, the fourth root is denoted as \( \sqrt[4]{x} \) or equivalently \( x^{1/4} \).
In the example provided, simplifying a fourth root includes breaking numbers into factorable components that can be simplified using the properties of exponents. The expression \( 125 \), written as \( 5^3 \), highlights how understanding base numbers helps simplify the fourth root: turning \( \sqrt[4]{125} \) into \( 5^{3/4} \).
Decomposing larger numbers like 125 into smaller, manageable factors simplifies radicals. By understanding this process, you can take complex-looking expressions and reduce them to their simple core, allowing for easy manipulation in algebraic equations.