Understanding the Square of -5 and Negative Numbers Raised to Powers

The square of a number involves multiplying the number by itself. When dealing with negative numbers, the result can be either positive or negative, depending on the exponent. Let's explore the square of -5 and negative numbers raised to different powers.

The Square of -5

The square of -5 is calculated by multiplying -5 by itself:

-5 x -5 25

Therefore, the square of -5 is 25. Squaring a negative number always results in a positive number because the product of two negative numbers is positive.

Raising a Number to a Power

Raising a number to a power is equivalent to multiplying that number by itself by as many times as the exponent specifies. For example, 3^2 means 3 x 3. When dealing with negative exponents, it is a common practice to enclose the negative number in parentheses to avoid ambiguity. For instance, -5^2 implies that -5 is multiplied by itself, which equals 25.

Here's a step-by-step breakdown:

Write out the expression using parentheses: (-5) * (-5) When multiplying two negative numbers, the result is positive: -5 * -5 25

Signs and Parentheses

Sometimes, the sign of a number can be critical, especially when dealing with exponents. The expression -5^2 might be misinterpreted as multiplying -1 by the square of 5. To avoid this, it's best to write the expression with parentheses: (-5)^2.

Here’s a typical scenario:

Write out the expression with parentheses: (-5) * (-5) When multiplying two negative numbers, the result is positive: -5 * -5 25

Therefore, the expression -5^2 results in 25, while -5 multiplied by -5 directly equals 25 as well.

Additional Examples and Conceptual Insights

Let's take a look at some additional examples:

Example 1: Squaring Negative Numbers

For an even power: -5^2 (-5) * (-5) 25 For an odd power: -5^3 (-5) * (-5) * (-5) -125

Squaring a negative number always results in a positive number, while raising a negative number to an odd power results in a negative number.

Example 2: Complex Expressions

Consider the following expressions:

5^2 5 * 5 25 -5^2 (-5) * (-5) 25

Both expressions result in 25, showing that the parentheses are crucial in determining the order of operations.

For complex expressions involving imaginary numbers, such as the square of the imaginary unit i (where i^2 -1):

square of -5 i * 5^(1/2) i * sqrt(5)

In this case, the square of -5 is considered as the square root of 5 multiplied by the imaginary unit i.

Conclusion

In conclusion, when dealing with negative numbers raised to powers, it is essential to be careful with signs and parentheses to avoid confusion. The square of -5 is 25, as shown by the examples and explanations provided.