Problem 29 - Elementary Algebra

Question

\((-10)^{2}\)

Step-by-Step Solution

Answer

100

Step 1: Identify the Base and Exponent

The given expression is \((-10)^2\). Here, -10 is the base and 2 is the exponent.

Step 2: Apply the Exponent

To simplify \((-10)^2\), you need to multiply the base (-10) by itself as many times as the exponent. In this case, you need to multiply -10 by itself 2 times.

Step 3: Calculate the Multiplication

Calculate \((-10) \times (-10) \). Multiplying two negative numbers results in a positive number. So the calculation is \(-10 \times -10 = 100\).

Step 4: Write Down the Final Answer

After performing the multiplication, the result is 100. Therefore, \((-10)^2 = 100\).

Key Concepts

Understanding Base and Exponent

In mathematics, exponents are used to represent repeated multiplication of the same number. An exponent consists of two parts: the 'base' and the 'exponent'. The base is the number being multiplied, and the exponent tells you how many times to multiply the base by itself.
For example, in the expression \((-10)^2\), \-10\ is the base and 2 is the exponent. This means \(-10\) is to be multiplied by itself two times, like this: \(-10 \times -10\).

Multiplying Negative Numbers

Multiplying negative numbers can sometimes seem confusing, but it's straightforward once you know the rules. When you multiply two negative numbers, the result is always a positive number.
This is because each negative sign essentially 'cancels out' the other.
For instance, if you multiply \(-10 \times -10\), you get 100.

• The first negative sign means 'opposite of 10'.
• The second negative sign turns it back around to 'opposite of the opposite of 10', which is positive 10.

So, \(-10 \times -10 = 100\).

Squaring a Number

When we 'square' a number, we are raising it to the power of two. This means multiplying the number by itself.
In the case of negative numbers, squaring them follows the same rule, but as we learned earlier, multiplying two negative numbers gives a positive result.
For example, if you square \(-10\), you calculate \(-10 \times -10 \). This gives us 100.

• Start by writing down the expression: \((-10)^2\)
• Next, calculate the multiplication: \(-10 \times -10\)
• The final result will be 100.

So, \((-10)^2 = 100\).