Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{63 a^{2}}-\sqrt{45 a^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \( 3a(\sqrt{7} - \sqrt{5}) \).

Step 1: Simplify the Individual Radicals

First, we simplify each square root individually. \( \sqrt{63a^2} \) can be written as \( \sqrt{63} \cdot \sqrt{a^2} \). Similarly, \( \sqrt{45a^2} = \sqrt{45} \cdot \sqrt{a^2} \). Since \( \sqrt{a^2} = a \), we have \( \sqrt{63a^2} = a \sqrt{63} \) and \( \sqrt{45a^2} = a \sqrt{45} \).

Step 2: Break Down Under-Root Numbers

Now, simplify \( \sqrt{63} \) and \( \sqrt{45} \) by factoring them. \( 63 = 9 \times 7 \) and \( 45 = 9 \times 5 \). Therefore, \( \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7} \) and \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \).

Step 3: Substitute and Combine Like Terms

Replace the simplified radicals back into the expression: \( a\sqrt{63} - a\sqrt{45} = a(3\sqrt{7}) - a(3\sqrt{5}) \). Factor out the common factor \( 3a \); the expression becomes \( 3a(\sqrt{7} - \sqrt{5}) \).

Step 4: Write the Final Simplified Expression

The expression \( 3a(\sqrt{7} - \sqrt{5}) \) is now in its simplest form since no further simplification is possible.

Key Concepts

Square Roots

Square roots are an essential part of understanding radical expressions. When you see the square root symbol (√), it represents a value that, when multiplied by itself, gives the original number under the root sign. For example, the square root of 9 is 3 because 3 times 3 equals 9.

When working with variables, like in our problem above, if you have a term such as \(\sqrt{a^2}\), you can use the property that the square root of a squared variable is simply the variable itself. This simplifies to \(a\) assuming \(a\) is non-negative, which is a common condition in algebra to avoid complications with imaginary numbers.

• The \(\sqrt{a^2} = a\) is applicable under the condition given that variables in the radicand with an even index are non-negative.
• If you have expressions such as \(\sqrt{63}\) or \(\sqrt{45}\), further simplification could be necessary to express them in terms of their simplest radical form.

Factoring Radicals

Factoring radicals helps simplify expressions, particularly when dealing with expressions involving numbers. The aim is to find perfect squares within the number under the square root, as this assists in breaking the number down into smaller parts.

In square roots like \(\sqrt{63}\) and \(\sqrt{45}\), we look for pairs of numbers that multiply to give you 63 and 45, respectively, highlighting any perfect squares. Perfect squares are numbers like 4, 9, 16, where the square root results in an integer.

• For \(\sqrt{63}\), you can rewrite this as \(9 \times 7\), where 9 is a perfect square. So \(\sqrt{63} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}\).
• Similarly, \(\sqrt{45}\) can be expressed as \(9 \times 5\), simplifiable to \(\sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\).
• These robust factorizations allow us to simplify expressions by pulling integers out of the square root, making further arithmetic simpler.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. It's crucial to simplify them to make computations more manageable. By simplifying, you can ensure algebraic expressions are easier to work with, especially when performing addition, subtraction, multiplication, or division.

In the expression \(a\sqrt{63} - a\sqrt{45}\), algebraic properties assist in simplifying. Algebra uses crucial steps, such as factoring, distributed throughout the expression, to manage it into a more workable form.

• First, factor any common elements. Here, \(a\) is common in both terms; hence, factor it out to start streamlining the expression.
• After factoring out \(a\), you're left with \(a(3\sqrt{7} - 3\sqrt{5})\). This indicates a further shared factor of 3, which can also be factored out to simplify enough to gain a final form: \(3a(\sqrt{7} - \sqrt{5})\).
• Combining like terms and simplifying using common factor rules is a powerful algebraic tool for reducing complex expressions, creating clarity and simplicity in mathematics.