How To Rewrite A Square Root

Rewriting square roots is a fundamental skill in algebra and beyond, allowing for simplification, easier manipulation, and clearer understanding of mathematical expressions. Mastering this process unlocks doors to solving complex equations, tackling intricate geometric problems, and simplifying seemingly daunting calculations. Let's dive into the art of rewriting square roots.

Understanding Square Roots

Before diving into the "how," let's establish a solid "what" and "why." A square root, denoted by the symbol √, asks a simple question: "What number, when multiplied by itself, equals the number under the radical?" This number under the radical is called the radicand.

• Perfect Squares: Numbers like 4, 9, 16, 25, and so on, are perfect squares because their square roots are whole numbers (2, 3, 4, 5, respectively).
• Non-Perfect Squares: Numbers like 2, 3, 5, 6, 7, etc., are non-perfect squares. Their square roots are irrational numbers, meaning they cannot be expressed as a simple fraction and their decimal representation goes on infinitely without repeating.

Rewriting square roots is essential because:

• Simplification: It allows us to express a square root in its simplest form, making it easier to work with.
• Combining Like Terms: Simplified square roots can be combined if they have the same radicand, similar to combining like variables in algebra.
• Solving Equations: Rewriting square roots is crucial for isolating variables and solving equations involving radicals.
• Rationalizing Denominators: It's a necessary step when we have a square root in the denominator of a fraction.

Core Principles of Rewriting Square Roots

The magic of rewriting square roots lies in understanding and applying these key principles:

• Product Property: The square root of a product is equal to the product of the square roots. Mathematically, √(ab) = √a * √b. This is the workhorse of simplification.
• Quotient Property: The square root of a quotient is equal to the quotient of the square roots. Mathematically, √(a/b) = √a / √b. This is useful for dealing with fractions under a radical.
• Identifying Perfect Square Factors: The ability to quickly recognize perfect square factors within the radicand is crucial. This allows us to "pull out" the square root of the perfect square, simplifying the overall expression.

Step-by-Step Guide to Rewriting Square Roots

Let's break down the process into a clear, step-by-step approach.

Step 1: Identify Perfect Square Factors

This is the most critical step. Look for the largest perfect square that divides evenly into the radicand. Here's how to do it effectively:

• List Perfect Squares: Mentally (or on paper) list perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.
• Divide and Conquer: Start dividing the radicand by these perfect squares, beginning with the largest. If it divides evenly (no remainder), you've found a perfect square factor.
• Example: Let's say we want to simplify √72.
  • Does 72 divide evenly by 100? No.
  • Does 72 divide evenly by 81? No.
  • Does 72 divide evenly by 64? No.
  • Does 72 divide evenly by 49? No.
  • Does 72 divide evenly by 36? Yes! 72 / 36 = 2. Therefore, 36 is a perfect square factor of 72.

Step 2: Apply the Product Property

Once you've identified a perfect square factor, rewrite the radicand as the product of the perfect square and its remaining factor. Then, apply the product property.

• Example (Continuing from above):
  • √72 = √(36 * 2)
  • √(36 * 2) = √36 * √2

Step 3: Simplify the Perfect Square Root

Take the square root of the perfect square factor.

• Example (Continuing from above):
  • √36 * √2 = 6 * √2
  • Therefore, √72 simplifies to 6√2

Step 4: Check for Further Simplification

Ensure that the remaining radicand (the number under the square root) has no further perfect square factors. If it does, repeat steps 1-3.

• Example: In our previous example, the radicand is 2. 2 has no perfect square factors other than 1, so we are done.

Let's work through more examples:

• Simplify √48:
  • Perfect square factor: 16 (48 / 16 = 3)
  • √48 = √(16 * 3) = √16 * √3 = 4√3
• Simplify √125:
  • Perfect square factor: 25 (125 / 25 = 5)
  • √125 = √(25 * 5) = √25 * √5 = 5√5
• Simplify √98:
  • Perfect square factor: 49 (98 / 49 = 2)
  • √98 = √(49 * 2) = √49 * √2 = 7√2

Rewriting Square Roots with Variables

The same principles apply when dealing with variables under the radical. The key is to remember the exponent rules.

• Rule: √(x²) = x (assuming x is non-negative). In general, √(x²ⁿ) = xⁿ where 'n' is an integer.
• Explanation: We're looking for pairs. x² means x * x, so we have a pair, and we can "pull out" one 'x'.

Steps:

1. Factor out Perfect Square Variable Terms: Identify variable terms with even exponents (x², x⁴, x⁶, etc.).
2. Apply the Product Property: Separate the perfect square variable terms from the remaining terms.
3. Simplify: Take the square root of the perfect square variable terms.

Examples:

• Simplify √(x³):
  • Rewrite: √(x² * x)
  • Apply product property: √x² * √x
  • Simplify: x√x
• Simplify √(16x⁵):
  • Rewrite: √(16 * x⁴ * x)
  • Apply product property: √16 * √x⁴ * √x
  • Simplify: 4x²√x
• Simplify √(25x²y³z⁴):
  • Rewrite: √(25 * x² * y² * y * z⁴)
  • Apply product property: √25 * √x² * √y² * √y * √z⁴
  • Simplify: 5xy z²√y

Rewriting Square Roots with Fractions

Dealing with fractions under a square root requires the quotient property and sometimes, rationalizing the denominator.

Step 1: Apply the Quotient Property

Separate the square root of the numerator and the square root of the denominator: √(a/b) = √a / √b.

Step 2: Simplify the Numerator and Denominator (if possible)

Simplify each square root independently, using the techniques described above.

Step 3: Rationalize the Denominator (if necessary)

Rationalizing the denominator means eliminating any square roots from the denominator. To do this, multiply both the numerator and denominator by the square root that's in the denominator.

Examples:

• Simplify √(9/16):
  • Apply quotient property: √9 / √16
  • Simplify: 3/4
• Simplify √(5/4):
  • Apply quotient property: √5 / √4
  • Simplify: √5 / 2
• Simplify √(3/7):
  • Apply quotient property: √3 / √7
  • Rationalize the denominator: (√3 / √7) * (√7 / √7) = (√3 * √7) / 7 = √21 / 7

Advanced Techniques and Considerations

• Nested Radicals: Sometimes, you might encounter radicals within radicals (e.g., √(4 + √7)). These are generally more complex and might require specialized techniques or might not be simplifiable in a neat way.
• Complex Numbers: When dealing with negative numbers under the square root, you enter the realm of complex numbers. Remember that √(-1) = i (the imaginary unit). For example, √(-9) = √(9 * -1) = √9 * √-1 = 3i.
• Higher Roots: The principles extend to cube roots, fourth roots, and so on. For cube roots, you're looking for perfect cube factors (8, 27, 64, etc.). For fourth roots, you're looking for perfect fourth power factors (16, 81, 256, etc.).

Common Mistakes to Avoid

• Incorrectly Applying the Product Property: √(a + b) ≠ √a + √b. The product property only applies to multiplication, not addition or subtraction.
• Forgetting to Simplify Completely: Always double-check that the remaining radicand has no further perfect square factors.
• Rationalizing Incorrectly: Make sure you multiply both the numerator and denominator by the same value when rationalizing.
• Ignoring Variable Restrictions: Technically, √(x²) = |x|, the absolute value of x. We often assume x is non-negative in introductory algebra to simplify things, but it's important to be aware of this nuance.

Practice Problems

Let's put your newfound knowledge to the test. Simplify the following square roots:

1. √50
2. √80
3. √(32x³)
4. √(18x⁴y⁵)
5. √(7/9)
6. √(2/5)
7. √(-16)

Solutions:

1. 5√2
2. 4√5
3. 4x√(2x)
4. 3x²y²√(2y)
5. √7 / 3
6. √10 / 5
7. 4i

Conclusion

Rewriting square roots is a powerful skill that builds a strong foundation for more advanced mathematical concepts. By understanding the core principles, practicing diligently, and avoiding common mistakes, you can master this art and unlock a new level of problem-solving ability. Remember to break down complex problems into smaller, manageable steps, and always double-check your work. Happy simplifying!