Problem 11 - Algebra 1

Question

Use the power of a power property to simplify the expression. $$ (-2)^{2} $$

Step-by-Step Solution

Answer

The result of \((-2)^2\) is \(4\).

Step 1: Understand the Power Rule

The power of a power property states that \((a^{m})^{n} = a^{m*n}\). In this case, \(-2\) is raised to the power \(2\), thus it will be \((-2)^2\).

Step 2: Apply the Power Rule

When you square anomaly number, regardless if it's negative or positive, the result is always positive. Therefore, \((-2)^2\) equals to \(4\).

Key Concepts

Exponentiation: Raising Numbers to a Power

Exponentiation is a mathematical operation that involves raising a number to a certain power. It is represented as \(a^n\), where \(a\) is the base and \(n\) is the exponent. The base is the number you want to multiply by itself and the exponent tells you how many times to do this multiplication.
For example, in the expression \((-2)^2\), \(-2\) is the base and \(2\) is the exponent. This means you multiply \(-2\) by itself once, resulting in -2 \(\times\) -2 = 4.

• Base: The number being multiplied.
• Exponent: The number of times the base is used as a factor.

In this special case since the exponent is 2, it's called squaring, which is just multiplying the number by itself. This is one of the simplest examples of exponentiation that every student should grasp effectively.

Simplifying Expressions Using Power Rules

Simplifying mathematical expressions helps to make them easier to work with and understand. The power rule, or the power of a power property, is a helpful tool in simplifying expressions involving exponents. The rule states that when raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m\cdot n}\).Using this rule can significantly simplify complex expressions.
In the exercise \((-2)^2\), the base \(-2\) is raised to the power of 2. Since there's only one exponent, the power rule reinforces that you compute the expression by multiplying the number by itself. Thus, simplifying this results in 4.

• Power Rule: Multiplies exponents in a power of a power expression.
• Reduction: Simplifies calculations by reducing steps required.

Remember, the power rule is a fundamental technique for simplifying such expressions, helping save both time and effort.

Integer Powers: Understanding Positive and Negative Bases

Integer powers involve exponents that are whole numbers and work similarly for both positive and negative bases. An integer exponent, say \(n\), means you need to multiply the base by itself \(n\) times. However, when dealing with negative bases, specific rules come into play to determine the sign of the result.When you raise a negative number to an even power, the result is positive, because negative multiplied by negative equals positive. In contrast, raising a negative number to an odd power results in a negative outcome.
In our example, \((-2)^2\) involves an even exponent of 2, making the product positive: 4.

• Even Powers: Result in positive outcomes when the base is negative.
• Odd Powers: Result in negative outcomes with negative bases.

Mastering the rules of integer powers is key to confidently handling expressions with negative bases. These concepts make evaluating expressions much more straightforward, especially in algebraic contexts.