Problem 6 - Algebra 2

Question

Simplify each expression. $$ 3^{\frac{1}{2}} \cdot 12^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(3^{1/2} \cdot 12^{1/2}\) is 6.

Step 1: Analyze the Given Expression

The given expression is \(3^{1/2} \cdot 12^{1/2}\). Here, \(1/2\) is the exponent which denotes square root operation.

Step 2: Convert Fractional Exponents to Roots

We convert the given expression into square root form. Therefore, it becomes \(\sqrt{3} \cdot \sqrt{12}\).

Step 3: Simplify the Expression

According to the rule of radicals, the multiplication of two square roots can be simplified into the square root of the multiplication of two numbers. Applying this rule, our expression becomes \(\sqrt{(3 \cdot 12)} = \sqrt{36}\).

Step 4: Solve the Square Root

The square root of 36 is 6. Therefore, the simplified form of the given expression is 6.

Key Concepts

Fractional Exponents

In mathematics, fractional exponents allow us to represent roots as powers. This concept is particularly useful when simplifying expressions involving radicals. A fractional exponent, like \(3^{\frac{1}{2}}\), means that the base is taken to the power of one half. This is equivalent to finding the square root of the base. Understanding fractional exponents helps with simplifying complex expressions by transforming roots into exponent form, which often makes calculations easier.

For example, \(a^{\frac{m}{n}}\) indicates the \(n\)-th root of \(a\) raised to the \(m\)-th power. When \(m=1\), it simply represents the \(n\)-th root of \(a\). Thus, \(3^{\frac{1}{2}}\) is the same as \(\sqrt{3}\).

Square Roots

Square roots are a fundamental concept in algebra and are closely tied to fractional exponents. The square root of a number \(x\) is a value that, when multiplied by itself, gives the original number \(x\). The square root symbol, \(\sqrt{}\), is commonly used to indicate this operation.

For instance, \(\sqrt{36} = 6\) because \(6 \times 6 = 36\). Understanding the connection between square roots and fractional exponents is crucial for algebraic simplifications because it allows us to move between different forms of mathematical expressions easily. It's often useful to replace a fractional exponent with its equivalent square root to make calculations more straightforward.

Radicals

Radicals are expressions that involve roots, such as square roots or cube roots. The term 'radical' comes from the Latin word "radix," which means "root." Radicals are frequently represented with the radical symbol \(\sqrt{}\) for square roots, but they can denote other roots with a small index number, like \(\sqrt[3]{}\) for cube roots.

Working with radicals involves several rules and properties, such as simplifying radicals by finding perfect square factors. For example, \(\sqrt{12}\) can be simplified to \(2\sqrt{3}\) by factoring 12 into 4 and 3, where 4 is a perfect square. Understanding how to manipulate radicals is essential for simplifying complex expressions in algebra.

Multiplication of Radicals

When dealing with radicals, multiplying expressions involves specific rules that simplify the process. According to the property of radicals, the product of two square roots is the square root of the product of the two numbers. This principle is crucial when simplifying expressions involving radicals and makes calculations more manageable.

For instance, when multiplying \(\sqrt{3} \cdot \sqrt{12}\), you can consolidate the two radicals into one: \(\sqrt{3 \times 12}\), which simplifies further into \(\sqrt{36}\). Since \(\sqrt{36} = 6\), the entire expression simplifies to 6. This process showcases how multiplication of radicals reduces complexity in expressions and is an essential skill for students to master when simplifying algebraic equations.