Problem 45 - Algebra 2

Question

Simplify. \((\sqrt{3}-\sqrt{5})^{2}\)

Step-by-Step Solution

Answer

The simplified form is \(8 - 2\sqrt{15}\).

Step 1: Apply the Binomial Square Formula

Recognize that you are dealing with a binomial square of the form \[(a - b)^2 = a^2 - 2ab + b^2.\]Here \(a = \sqrt{3}\) and \(b = \sqrt{5}\). Substitute these into the formula to start simplifying.

Step 2: Calculate the Squares of Each Term

Calculate \(a^2 = (\sqrt{3})^2 = 3\) and \(b^2 = (\sqrt{5})^2 = 5\). Write these down for use in the next step.

Step 3: Multiply the Middle Term

Find \(2ab = 2 \cdot \sqrt{3} \cdot \sqrt{5}\). Use the property \(\sqrt{x}\sqrt{y} = \sqrt{xy}\) to simplify this to \(2 \cdot \sqrt{15}\).

Step 4: Combine All Terms

Substitute all calculated parts back into the binomial square formula: \[(\sqrt{3} - \sqrt{5})^2 = a^2 - 2ab + b^2 = 3 - 2\sqrt{15} + 5.\]

Step 5: Perform Final Simplification

Combine the constant terms: \[3 + 5 = 8.\]Therefore, the expression simplifies to \[8 - 2\sqrt{15}.\]

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic expressions into their simplest form. This makes them easier to work with or recognize, especially when solving equations or performing further mathematical operations. In our exercise, we had the binomial expression

• \((\sqrt{3}-\sqrt{5})^{2}\).

To simplify this, we applied a standard algebraic technique known as the 'binomial square formula'. This formula helps you expand expressions of the form \((a-b)^2\) into an easily calculable format:

• \(a^2 - 2ab + b^2\).

This transformation allows us to work with individual squares and the product of terms, step by step, eventually leading to an expression that is simple and straightforward.

Square Roots

Square roots are a fundamental concept in algebra, representing a value that, when multiplied by itself, gives the original number. For example,

• \(\sqrt{3}\) is a number which, if multiplied by itself, results in 3.
• Similarly, \(\sqrt{5}\) squared equals 5.

In the exercise, both \(a\) and \(b\) involve square roots of numbers. Recognizing square roots allows you to calculate the term \(a^2\) as \((\sqrt{3})^2\), simplifying directly to 3, and \(b^2 = (\sqrt{5})^2 = 5\). Understanding square roots is crucial for manipulations involving real numbers and preparing for more advanced mathematical topics.

Algebraic Formulas

Algebraic formulas are powerful tools that give you shortcuts to solve problems involving variables. One such formula, the binomial square formula, allows you to expand binomial expressions quickly. It is written as:

• \((a-b)^2 = a^2 - 2ab + b^2\).

Knowing this formula helps, as it transforms expressions like \((\sqrt{3} - \sqrt{5})^2\) into something manageable.
Once decomposed using the formula, we found:

• \(a^2 = 3\)
• \(b^2 = 5\)
• \(2ab = 2 \times \sqrt{3} \times \sqrt{5} = 2\sqrt{15}\)

Combining these helped us simplify the expression to

• \(8 - 2\sqrt{15}\).

Mastering algebraic formulas not only saves time but also helps increase precision in solving many different types of algebraic problems.