Problem 75 - Algebra and Trigonometry

Question

Let \(a, b,\) and \(c\) be real numbers with \(a > 0, b < 0,\) and \( c < 0 .\) Determine the sign of each expression. \(\begin{array}{ll}{\text { (a) } b^{5}} & {\text { (b) } b^{10} \quad \text { (c) } a b^{2} c^{3}} \\ {\text { (d) }(b-a)^{3}} & {\text { (e) }(b-a)^{4}} \quad {\text { (f) } \frac{a^{3} c^{3}}{b^{6} c^{6}}}\end{array}\)

Step-by-Step Solution

Answer

(a) negative, (b) positive, (c) negative, (d) negative, (e) positive, (f) negative.

Step 1: Analyzing Expression (a)

The expression is \(b^5\). Since \(b < 0\), raising a negative number to an odd power gives a negative result. Thus, \(b^5 < 0\).

Step 2: Analyzing Expression (b)

The expression is \(b^{10}\). Since \(b < 0\), raising a negative number to an even power gives a positive result. Thus, \(b^{10} > 0\).

Step 3: Analyzing Expression (c)

The expression is \(ab^{2}c^{3}\). We know \(a > 0\), \(b < 0\), and \(c < 0\). Raising \(b\) to an even power \(b^2\) makes it positive. Raising \(c\) to an odd power \(c^3\) keeps it negative. Hence, the product \(ab^2c^3\) is positive \(\times\) positive \(\times\) negative, resulting in a negative value. Thus, \(ab^2c^3 < 0\).

Step 4: Analyzing Expression (d)

The expression is \((b-a)^3\). We know \(b < 0\) and \(a > 0\) thus the term \(b-a = b + (-a) < 0\). Raising a negative number to an odd power results in a negative number. Thus, \((b-a)^3 < 0\).

Step 5: Analyzing Expression (e)

The expression is \((b-a)^4\). Since \(b-a < 0\) as shown previously, raising a negative number to an even power results in a positive number. Thus, \((b-a)^4 > 0\).

Step 6: Analyzing Expression (f)

The expression is \(\frac{a^{3}c^{3}}{b^{6}c^{6}}\). The numerator \(a^3 c^3\) results in a positive \(\times\) negative = negative. The denominator \(b^6 c^6\) results in positive because \(b\) and \(c\) raised to an even power both yield positive values. Thus, the fraction is negative over positive, resulting in \(\frac{a^3 c^3}{b^6 c^6} < 0\).

Key Concepts

Real Numbers Properties

Real numbers are the foundation of many concepts in algebra and mathematics. They include both rational and irrational numbers. Rational numbers can be expressed as a fraction of two integers, like 1/2 or -3. Irrational numbers cannot be expressed as simple fractions, such as π or √2. Some key properties of real numbers include:

• Closure: Real numbers are closed under addition, subtraction, multiplication, and division (except by zero).
• Commutativity: The order in which numbers are added or multiplied does not affect the result, i.e., a + b = b + a and ab = ba.
• Associativity: When adding or multiplying, the way in which the numbers are grouped does not affect the outcome, i.e., (a + b) + c = a + (b + c).
• Distributive Property: Multiplication distributes over addition, i.e., a(b + c) = ab + ac.

Understanding these properties helps in evaluating and simplifying expressions involving real numbers. It's also essential for recognizing how the signs of numbers change with different operations.

Powers of Numbers

In mathematics, raising numbers to powers is a crucial operation. The power or exponent of a number indicates how many times the number is multiplied by itself. The base is the number being multiplied, and the exponent is the small raised number indicating repetition. For instance, in 5³, 5 is the base and 3 is the exponent.

• Odd Exponents: When a negative number is raised to an odd power, the result is negative. For example, (-2)³ = -8.
• Even Exponents: Raising a negative number to an even power results in a positive number. For example, (-3)² = 9.
• Zero as an Exponent: Any non-zero number raised to the power of zero equals one. For instance, 7⁰ = 1.

Recognizing these patterns aids in predicting whether an expression yields a positive or negative result depending on the exponent's parity and the base's sign.

Negative and Positive Multiplication

Understanding how signs change in multiplication is fundamental in algebra and real numbers. When multiplying two numbers, the sign of the result depends on the signs of the numbers:

• Positive × Positive = Positive: Multiplying two positive numbers results in a positive number, such as 3 × 4 = 12.
• Negative × Negative = Positive: A negative times a negative equals a positive, such as (-2) × (-3) = 6.
• Positive × Negative = Negative: A positive times a negative results in a negative, like 5 × (-1) = -5.
• Negative × Positive = Negative: Similarly, a negative times a positive is negative, e.g., (-4) × 7 = -28.

These rules help determine the overall sign of a product in complex expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They often represent real-life quantities or serve as abstract mathematical tools. Here's a brief breakdown:

• Variables: Symbols like x, y, or z that represent numbers. The specific value isn't given, allowing for generalization.
• Constants: Known values in the expression, like 3 or -5.
• Coefficients: Numbers multiplying the variables, indicating how many times the variable will be added together.
• Terms: Parts of the expression separated by addition or subtraction, like in 3x + 4.

Solving algebraic expressions involves combining like terms, using distributive properties, and understanding the operations' order. It's crucial to know how each element of the expression influences the overall value and structure.