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Chapter 2: Problem 55

Compute the exact value. \((-6)^{2}\)

Short Answer

Expert verified
36

Step by step solution

01

Identify the Base and Exponent

In the expression \((-6)^{2}\), the number inside the parentheses, -6, is the base and the number outside, 2, is the exponent. The expression represents -6 multiplied by itself.
02

Apply the Exponent

Since the exponent is 2, we need to multiply the base, -6, by itself. This means performing the operation -6 脳 -6.
03

Perform Multiplication

Multiply -6 by -6. Multiplying two negative numbers results in a positive number, so (-6) 脳 (-6) = 36.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a way to express how many times a number, known as the base, is multiplied by itself. In our example, \((-6)^{2}\), the exponent is 2. This means we are multiplying the base, which is -6, by itself. Exponents are useful because they simplify repeated multiplication. Rather than writing \(-6 \times -6\), we use \((-6)^{2}\) to show the operation concisely.
  • **Base:** The number being multiplied.
  • **Exponent:** Indicates how many times the base multiplies itself.
Exponents come with certain rules, such as the product of powers rule \((a^{m} \times a^{n} = a^{m+n})\) and the power of a power rule \(( (a^{m})^{n} = a^{m \cdot n} )\). Understanding these rules can help when working with more complex expressions.
Negative Numbers
Negative numbers are values less than zero and are often represented with a minus sign. They have particular rules when used in operations like addition, subtraction, multiplication, and division.
  • When adding two negative numbers, the result is more negative.
  • When subtracting, consider the direction: subtracting a larger negative from a smaller one yields a positive result.
  • Multiplying two negative numbers results in a positive number.
  • Dividing two negative numbers also results in a positive number.
When multiplying -6 by itself, as in \((-6) \times (-6)\), the negatives cancel out, resulting in a positive product. This is crucial in understanding how negative signs work during multiplication.
Multiplication
Multiplication refers to repeated addition. For example, instead of adding -6 twice, we multiply to simplify the operation: \(-6 + (-6) = -12\) becomes \(-6 \times 2\) or \((-6)^{2}\). Multiplying safeguards us from lengthy and error-prone addition.
Multiplication has its own set of rules:
  • **Commutative Property:** \(a \times b = b \times a\)
  • **Associative Property:** \((a \times b) \times c = a \times (b \times c)\)
  • **Distributive Property:** \(a \times (b + c) = a \times b + a \times c\)
In our exercise, we use \(-6 \times -6\) to emphasize multiplying negatives leads to positive results, reinforcing the sign rules and basic arithmetic guidelines.

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