Problem 60 - Algebra 1

Question

Which is the simplificd form of \(\frac{6 \sqrt{52}}{\sqrt{2}} \cdot \sqrt{8}^{?}\) (A) \(\frac{3 \sqrt{13}}{\sqrt{2}}\) (B) \(3 \sqrt{13}\) (C) \(3 \sqrt{26}\) (D) \(\frac{6 \sqrt{13}}{\sqrt{2}}\)

Step-by-Step Solution

Answer

The given alternatives are incorrect as none of them matches the final simplified form of the expression, i.e., \(24 \sqrt{26}\).

Step 1: Simplify the radicals

Start by simplifying each radical. Remember that \(\sqrt{a*b} = \sqrt{a} * \sqrt{b}\), and \(\sqrt{a^2} = a\): \[\frac{6 \sqrt{52}}{\sqrt{2}} \cdot \sqrt{8} = \frac{6 \sqrt{4*13}}{\sqrt{2}} \cdot \sqrt{4*2} = \frac{6*2\sqrt{13}}{\sqrt{2}} \cdot 2\sqrt{2}\]

Step 2: Cancel out common terms

Now, cancel out the common terms between numerator and denominator, and simplify multiplication: \[= \frac{12\sqrt{13}}{1} \cdot 2\sqrt{2} = 12\sqrt{13} \cdot 2\sqrt{2}\]

Step 3: Simplify the expression

Finally, we simplify the expression by multiplying the terms: \[= 24\sqrt{26}\]

Step 4: Match the simplified expression to the alternatives

Now, compare the simplified expression to the given alternatives. None of them matches with the final simplified form, so there seems to be an error in the alternatives given.

Key Concepts

Radical Expressions

Understanding radical expressions is essential for simplifying complex numbers that involve roots. A radical expression can be written in the form \( \sqrt[n]{x} \) where \( x \) is the radicand (the number under the root) and \( n \) is the index indicating the degree of the root. In most cases, we deal with square roots where the index \( n \) equals 2, but roots can be of any degree.

When simplifying radicals, it's important to identify and factor out perfect squares, cubes, or higher powers, depending on the root's index. For instance, \( \sqrt{52} \) can be split into \( \sqrt{4 \times 13} \) because 4 is a perfect square, making the simplification process easier by extracting \( \sqrt{4} \) as 2 from under the radical, resulting in \( 2\sqrt{13}\).

• Identify perfect squares, cubes, etc., within radicands to break them down.
• Remember that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
• Apply the simplification outside the radical to reduce the expression further.

Square Roots

Square roots are a type of radical expression with an index of 2, written as \( \sqrt{x} \) where \( x \) is the radicand. Simplifying a square root involves finding the largest perfect square factor of the radicand and then taking the square root of that factor out of the radical.

For example, \( \sqrt{8} \) can be simplified since 8 is \( 4 \times 2 \) and 4 is a perfect square. This lets us rewrite \( \sqrt{8} \) as \( \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \) because \( \sqrt{4} \) equals 2.

• Finding the largest perfect square factor simplifies the expression.
• Remember that multiplying a square root by itself gives the radicand (e.g., \( \sqrt{2} \cdot \sqrt{2} = 2 \) ).
• Always look to simplify before performing operations to make the process easier.

Arithmetic Operations

Combining radical expressions often requires the use of basic arithmetic operations: addition, subtraction, multiplication, and division. When multiplying radicals, you can multiply the radicands together if the indices of the roots are equal. Division, on the other hand, may require rationalizing the denominator, which means getting rid of the radical in the denominator.

As seen in the exercise example, \( \frac{6 \sqrt{52}}{\sqrt{2}} \cdot \sqrt{8} \) simplifies to \( \frac{12\sqrt{13}}{1} \cdot 2\sqrt{2} = 24\sqrt{26} \) after the multiplication of the terms. Key things to keep in mind include:

• Multiplication of radicals: Combine radicands and simplify.
• Division of radicals: Rationalize the denominator, if needed.
• Only combine like terms (same radicand and index).

When simplifying arise, it’s essential to apply these arithmetic operations accurately to avoid errors and match the correct answer.