Problem 67 - Algebra 2

Question

Simplify each expression. \(6^{\frac{1}{2}} \cdot 12^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(6*\sqrt{2}\)

Step 1: Simplify \(6^{\frac{1}{2}}\)

The exponent \(\frac{1}{2}\) implies taking square root. So, \(6^{\frac{1}{2}}\) equals to \(\sqrt{6}\).

Step 2: Simplify \(12^{\frac{1}{2}}\)

In a similar way, \(12^{\frac{1}{2}}\) equals to \(\sqrt{12}\).

Step 3: Multiply the results

Multiply the results we got from steps 1 and 2 as follows: \(\sqrt{6} * \sqrt{12}\)

Step 4: Simplify the result

According to multiplication rule of radicals, we can combine the two values under the same square root: \(\sqrt{6 * 12}\)

Step 5: Calculate the Product

Multiply 6 and 12 to get 72: So, we have \(\sqrt{72}\)

Step 6: Simplify the square root

The square root of 72 can be simplified. The largest square number that divides into 72 is 36. Therefore, we have \(\sqrt{72} = \sqrt{36*2} = 6*\sqrt{2}\)

Key Concepts

Square roots

Square roots are a fundamental concept in mathematics, often represented by the radical symbol (√). The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

Square roots are integral in simplifying expressions involving exponents, especially when dealing with fractional exponents like \( x^{\frac{1}{2}} \). This notation simply means "the square root of x." For example, \( 6^{\frac{1}{2}} \) is the same as \( \sqrt{6} \). Breaking down expressions in this manner helps in simplifying and solving complex equations.

Radicals

Radicals extend the idea of roots beyond just squares, encompassing cube roots, fourth roots, and so on. However, in most basic applications, radicals refer to square roots.

When you encounter a number with \( x^{1/n} \), it is read as the nth root of \( x \) and represented as \( \sqrt[n]{x} \). Radicals are important for solving problems where values need to be broken down or simplified within root expressions.

• For instance, \( \sqrt{6} \) and \( \sqrt{12} \) are both examples of radicals, each representing the principal square root of their respective numbers.

This is a common technique used to simplify or combine terms under a single radical expression, as you can often factor expressions to make them easier to manipulate.

Simplifying expressions

Simplifying expressions involves reducing them to their simplest form. This often entails removing any unnecessary parts or making the expression easier to understand and use.

When you have equivalent exponents or roots, like in our problem, simplifying means you might need to rewrite them under a common radical, which is consistently easier to manage.

• In the exercise, simplifying \( \sqrt{72} \) to \( 6\sqrt{2} \) is an example of this. You find the largest perfect square factor of 72, which is 36, and use it to break down \( \sqrt{72} \) into \( \sqrt{36} \cdot \sqrt{2} \), simplifying further to \( 6\sqrt{2} \).

This process makes equations more accessible for further calculation or interpretation, a crucial skill in mastering mathematics.

Multiplication of radicals

Multiplying radicals involves using the multiplication property of square roots, which states that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \). This property allows you to combine two separate radical expressions into one.

For example, in the exercise, multiplying \( \sqrt{6} \) by \( \sqrt{12} \) gives us \( \sqrt{72} \). This is because you can multiply the numbers inside the radicals first (\( 6 \times 12 = 72 \)) and then take the square root of the product.

• This rule simplifies the problem by reducing the number of separate calculations you need to make and helps simplify radical expressions effectively.

Therefore, understanding and applying this property is key to working with and simplifying more complex mathematical equations involving radicals.