GMAT Online Course: Demystifying Square Roots

Square roots are the inverse operation of squaring a number, and they're a common feature on the GMAT. Understanding them thoroughly is crucial for algebra, number properties, and even some geometry problems. Let's break down the essentials!

What is a Square Root?

At its simplest, finding the square root of a number 'n' means finding a number 'x' that, when multiplied by itself, equals 'n'.

• Definition: If x^2 = n, then x is a square root of n.

• Notation: The symbol for a square root is the radical sign, sqrt(). So, sqrt(n) means "the square root of n".

Positive and Negative Square Roots

For any positive number 'n', there are actually two real square roots: one positive and one negative.

• Example: For the number 9:

  • 3^2 = 9

  • (-3)^2 = 9 So, both 3 and -3 are square roots of 9.

• The Principal Square Root: When you see the radical sign sqrt(n) on its own, it refers only to the principal (positive) square root.

  • sqrt(9) = 3 (Always the positive root)

  • -sqrt(9) = -3 (The negative of the principal root)

  • If a problem asks for "all square roots" or gives you x^2 = 9, then the answer is x = +/-3. But if it just shows sqrt(9), the answer is only 3. This is a critical distinction for the GMAT!

The Absolute Value Property: sqrt(x^2) = |x|

This is a frequently tested concept! When you take the square root of a squared variable, the result is the absolute value of that variable.

• Rule: sqrt(x^2) = |x|

• Why? Because the square root symbol (sqrt()) always denotes the principal (non-negative) square root.

  • If x = 3, then sqrt(3^2) = sqrt(9) = 3. Here, |x| = |3| = 3. Matches!

  • If x = -3, then sqrt((-3)^2) = sqrt(9) = 3. Here, |x| = |-3| = 3. Matches! If you just wrote sqrt(x^2) = x, it would be wrong when x is negative. Hence, the absolute value is necessary to guarantee a non-negative result.

What about Square Roots of Negative Numbers?

• Can you square a real number and get a negative result? No. 3^2 = 9 and (-3)^2 = 9.

• Therefore, the square root of a negative number (e.g., sqrt(-4)) is not a real number. These are called imaginary numbers.

• On the GMAT, you will almost exclusively deal with real numbers. If you encounter a square root of a negative number, it usually indicates that the expression is undefined in the real number system or that no real solution exists.

Why Square Roots Matter for the GMAT:

• Simplification: Simplifying radical expressions.

• Solving Equations: Solving quadratic equations (e.g., x^2 = 25).

• Number Properties: Understanding properties of perfect squares.

• Absolute Value: Directly ties into absolute value concepts for variables.

Mastering these core ideas about square roots will significantly improve your performance on related GMAT problems. Pay special attention to the principal root and the absolute value property!

Version 2: Self-Paced Course - "Lesson 3.3: Square Roots – Unpacking the Inverse of Squaring"

Welcome to Lesson 3.3! After exploring the intricacies of exponents, we now turn our attention to their direct inverse operation: square roots. Just as addition undoes subtraction and multiplication undoes division, finding a square root undoes the operation of squaring a number. Square roots are ubiquitous on the GMAT, appearing in various contexts from basic arithmetic and number properties to algebraic equations and geometry problems (think Pythagorean theorem!). A thorough understanding of their definition, properties, and nuances is indispensable for achieving accuracy and efficiency on the exam.

This lesson will provide a comprehensive examination of square roots, covering their fundamental definition, the critical distinction between positive and negative roots, the absolute value property (a common GMAT trap!), and a brief but important introduction to the concept of imaginary numbers.

I. Definition of a Square Root: The Quest for the Original Number

At its core, a square root answers the question: "What number, when multiplied by itself, gives me this original number?"

• Formal Definition: A number 'x' is a square root of a number 'n' if, when 'x' is squared (multiplied by itself), the result is 'n'. In mathematical terms:

  If x^2 = n, then x is a square root of n.

• Notation: The primary symbol used to denote a square root is the radical sign, written as sqrt().

  • sqrt(n) means "the square root of n."

  • The number under the radical sign (n) is called the radicand.

• Perfect Squares: Numbers whose square roots are integers are called perfect squares.

  • sqrt(4) = 2 (because 2^2 = 4)

  • sqrt(81) = 9 (because 9^2 = 81)

  • sqrt(144) = 12 (because 12^2 = 144)

• Non-Perfect Squares: Numbers whose square roots are not integers are irrational numbers.

  • sqrt(2) approx 1.414...

  • sqrt(7) approx 2.646... On the GMAT, you often leave these in radical form unless asked to approximate.

II. Positive and Negative Square Roots: The Dual Nature

This is one of the most critical distinctions for the GMAT. For any positive number 'n', there are actually two real numbers that, when squared, result in 'n'.

• The Principle:

  • For a positive number, say 25:

    • 5^2 = 25

    • (-5)^2 = 25 Therefore, both 5 and -5 are square roots of 25.

• The Principal (Positive) Square Root: When the radical symbol sqrt() is used alone (without a positive or negative sign in front of it), it always refers to the principal (non-negative) square root.

  • sqrt(36) = 6 (It is always the positive value, by definition)

  • sqrt(0.09) = 0.3

  • sqrt(1/4) = 1/2

• Denoting Both Roots:

  • If you want to refer to only the negative square root, you place a negative sign outside the radical:

    • -sqrt(36) = -6

  • If you want to refer to both the positive and negative square roots, you use the +/- symbol:

    • +/-sqrt(36) = +/-6 (This means both +6 and -6)

• GMAT Implications - Solving Equations: This distinction is crucial when solving equations involving squares.

  • If a question asks for sqrt(9), the answer is only 3.

  • If a question gives you x^2 = 9 and asks for x, the answer is x = +/-3. You must consider both possibilities because 'x' could be either positive or negative. Forgetting the negative root is a common error on the GMAT.

III. The Absolute Value Property: sqrt(x^2) = |x|

This property is a direct consequence of the definition of the principal square root and is very frequently tested, especially when variables are involved.

• Rule: For any real number x, sqrt(x^2) = |x|.

  The result of a square root operation (specifically the principal square root) must always be non-negative.

• Why is this necessary?

  Let's test with examples:

  • If x = 5: sqrt(x^2) = sqrt(5^2) = sqrt(25) = 5. Here, |x| = |5| = 5. (Matches)

  • If x = -5: sqrt(x^2) = sqrt((-5)^2) = sqrt(25) = 5. Here, |x| = |-5| = 5. (Matches)

  Notice that if you simply wrote sqrt(x^2) = x, it would be incorrect when x is negative (e.g., if x = -5, sqrt((-5)^2) equals 5, not -5). The absolute value ensures that the output of sqrt(x^2) is always the positive (principal) root, aligning with the definition of the square root symbol.

• GMAT Relevance:

  • This property is especially important in Data Sufficiency questions where variables are involved, and you need to determine the value or range of a variable.

  • It also appears in algebra problems where simplifying expressions with variables under a square root is required.

  • Example: If sqrt(y^4) = 16, what is y? sqrt((y^2)^2) = |y^2|. Since y^2 is always non-negative, |y^2| = y^2. So, y^2 = 16, which means y = +/-4.

IV. Introduction to Imaginary Numbers: When Square Roots Break Real Rules

Up to this point, all numbers we've discussed are real numbers. However, not all numbers have real square roots.

• The Problem: Consider sqrt(-4). Can you think of a real number that, when multiplied by itself, results in -4?

  • 2 * 2 = 4

  • (-2) * (-2) = 4 There is no real number that, when squared, yields a negative result.

• Definition of 'i' (Imaginary Unit): To address this, mathematicians defined an imaginary unit, denoted by 'i'.

  • i = sqrt(-1)

  • By definition, i^2 = -1

• Imaginary Numbers: A number that is the square root of a negative number is called an imaginary number.

  • sqrt(-4) = sqrt(4 -1) = sqrt(4) sqrt(-1) = 2i

  • sqrt(-9) = sqrt(9 -1) = sqrt(9) sqrt(-1) = 3i

  • sqrt(-25) = 5i

• GMAT Relevance:

  • The GMAT primarily deals with real numbers. Unless a problem explicitly mentions "complex numbers" or "imaginary numbers" (which is rare to non-existent for the standard GMAT Quant section), you should assume all variables and results are within the real number system.

  • If you encounter sqrt(-n) where n is positive, and the problem expects a real number answer, it typically means the expression is undefined in the context of real numbers or that no real solution exists. This might be an answer choice for a "which of the following is defined" type of question.

V. Why Understanding Square Roots Matters for the GMAT

• Solving Quadratic Equations: Square roots are essential for solving equations of the form x^2 = n, which are very common.

• Simplifying Radical Expressions: Knowledge of square roots helps in simplifying expressions involving radicals, often required to match answer choices.

• Number Properties: Understanding perfect squares, non-perfect squares, and properties like units digits of squares.

• Algebraic Manipulation: Manipulating expressions containing variables under square roots requires precise application of rules, especially the absolute value property.

• Geometry Applications: The Pythagorean theorem (a^2 + b^2 = c^2) frequently leads to calculations involving square roots.

• Data Sufficiency: Questions often hinge on determining whether a variable is positive, negative, or zero, particularly when dealing with squared terms or square roots.

VI. Common Pitfalls to Avoid

• Forgetting the Negative Root: When solving x^2 = n, always remember that x = +/-sqrt(n). This is probably the most frequent square root mistake.

• Ignoring the Absolute Value: Forgetting that sqrt(x^2) = |x| is a critical error when x can be negative. Never assume sqrt(x^2) = x unless you are explicitly told x >= 0.

• Distributing Roots Over Addition/Subtraction: sqrt(a + b) is NOT sqrt(a) + sqrt(b). For example, sqrt(9+16) = sqrt(25) = 5, but sqrt(9) + sqrt(16) = 3 + 4 = 7. These are not equal. This applies to subtraction as well.

• Misconception about sqrt(n)'s Sign: Always remember that sqrt(n) by itself means ONLY the non-negative (principal) root.

• Operating with Imaginary Numbers: Unless explicitly stated, GMAT problems assume real numbers. sqrt(-number) is not a real number.

VII. Interactive Check Your Understanding:

1. What are all the real square roots of 121?

2. What is the value of -sqrt(64)?

3. Simplify sqrt((-7)^2).

4. If y^2 = 49, what are the possible values of y?

5. Is sqrt(-16) a real number? Explain.

VIII. Practice Questions:

1. If x^2 = 169, which of the following could be the value of x?

   a) -13

   b) 13

   c) Both a and b

   d) Neither a nor b

2. Simplify: sqrt(49) - sqrt((-7)^2)

3. If x is a real number, and sqrt(x^2) = 10, what are the possible values of x?

   a) 10 only

   b) -10 only

   c) 10 and -10

   d) No real solution

4. Which of the following expressions is NOT a real number?

   a) sqrt(1/4)

   b) -sqrt(25)

   c) sqrt(0)

   d) sqrt(-0.01)

5. Given that a = 3 and b = -5, evaluate sqrt(a^2) + sqrt(b^2).

Solutions to Practice Questions:

1. If x^2 = 169, which of the following could be the value of x?

   When x^2 = n, x = +/-sqrt(n).

   So, x = +/-sqrt(169).

   Since 13^2 = 169, x = +/-13.

   Both 13 and -13 are possible values.

   The correct answer is c) Both a and b.

2. Simplify: sqrt(49) - sqrt((-7)^2)

   • sqrt(49) = 7 (This is the principal, positive root)

   • sqrt((-7)^2) = sqrt(49) = 7 (Using sqrt(x^2) = |x|, so sqrt((-7)^2) = |-7| = 7)

   • So, 7 - 7 = 0. The value is 0.

3. If x is a real number, and sqrt(x^2) = 10, what are the possible values of x?

   Using the absolute value property, sqrt(x^2) = |x|.

   So, |x| = 10.

   This means x can be 10 or x can be -10, because both |10| = 10 and |-10| = 10.

   The correct answer is c) 10 and -10.

4. Which of the following expressions is NOT a real number?

   a) sqrt(1/4) = 1/2 (Real number)

   b) -sqrt(25) = -5 (Real number)

   c) sqrt(0) = 0 (Real number)

   d) sqrt(-0.01) = sqrt(-1 0.01) = sqrt(-1) sqrt(0.01) = i * 0.1 = 0.1i (Imaginary number, NOT a real number)

   The correct answer is d) sqrt(-0.01).

5. Given that a = 3 and b = -5, evaluate sqrt(a^2) + sqrt(b^2).

   • sqrt(a^2) = sqrt(3^2) = sqrt(9) = 3 (Since a is positive, |a| = a)

   • sqrt(b^2) = sqrt((-5)^2) = sqrt(25) = 5 (Using sqrt(x^2) = |x|, so |-5| = 5)

   • So, 3 + 5 = 8. The value is 8.