Problem 50 - Beginning and Intermediate Algebra

Question

Evaluate. $$ (-7)^{2} $$

Step-by-Step Solution

Answer

49

Step 1: Understanding the Problem

We need to evaluate the expression \((-7)^2\). This means we need to multiply -7 by itself.

Step 2: Performing the Multiplication

Multiply -7 by -7. We have: \[ (-7) \times (-7) = 49 \] When multiplying two negative numbers, the product is positive.

Step 3: Conclusion

After performing the multiplication, we have determined that \((-7)^2 = 49\).

Key Concepts

Exponentiation

Exponentiation is a fundamental part of algebra that involves raising a number to a power. In simple terms, this means multiplying the number by itself as many times as the exponent indicates.

For example, in a^b , the number a is called the base, and b is the exponent or power. If you have (-7)^2 , the base is -7 and the exponent is 2 . The expression (-7)^2 reads as "negative seven squared", and indicates that -7 should be multiplied by itself.

When solving problems involving exponents, remember:

• Any number raised to the power of 1 is itself.
• Any number raised to the power of 0 is 1, except if the base is also 0.
• When an exponent is 2, it often describes a square in geometry, giving rise to the term "squared."

To solve an expression like (-7)^2 , simply multiply -7 by itself, resulting in 49 . This process showcases how exponentiation operates in algebra.

Negative numbers

Negative numbers can be a tricky concept, but they are crucial in algebra. A negative number is less than zero and is usually written with a minus sign in front of it, such as -7 .

Negative numbers are often used to denote a reversal of quantity, like a debit in finance or a drop in temperature. A deep understanding of how to work with negative numbers can make solving algebraic expressions much easier.

When working with negative numbers:

• Adding two negative numbers results in a more negative number, i.e. -3 + (-2) = -5 .
• Subtracting a negative is like adding the number's absolute value, i.e. 5 - (-3) = 8 because you add 3 to 5 .
• Multiplying two negative numbers gives a positive result, such as -7 imes -7 = 49 . This is because the two negative signs "cancel out."

Remember, understanding how to handle negative numbers opens doors to solving more advanced algebra problems with confidence.

Multiplication rules

In algebra, multiplication rules dictate how numbers interact when multiplied together. These rules are essential when dealing with negative numbers, as demonstrated in expressions like (-7)^2 .

Multiplying numbers involves taking a number and adding it to itself multiple times. For example, 4 imes 3 means adding 4 together three times (i.e., 4 + 4 + 4 = 12 ).

When multiplying, keep these rules in mind:

• Positive × Positive = Positive: For example, 3 imes 2 = 6 .
• Negative × Positive = Negative: For example, -5 imes 2 = -10 .
• Negative × Negative = Positive: As seen with -7 imes -7 = 49 , multiplying two negatives results in a positive outcome.

By mastering these multiplication rules, including how they apply to negative numbers, solving algebraic expressions becomes straightforward. Remember that the order in which numbers are multiplied does not affect the result, thanks to the commutative property of multiplication.