Problem 13 - Beginning and Intermediate Algebra

Question

Simplify. $$m^{-5} \cdot m^{5}$$

Step-by-Step Solution

Answer

The simplified expression is \(m^{0}\), which equals 1.

Step 1: Identifying the exponent properties

We need to apply the exponent multiplication property which states that when multiplying the same base (in this case, m) with different exponents, you simply add the exponents together. So we have: \(m^{-5} \cdot m^{5}\)

Step 2: Applying the exponent multiplication property

Now, apply the exponent multiplication property to the expression: \(m^{(-5+5)}\)

Step 3: Simplify the expression

Finally, simplify the expression by adding the exponents: \(m^{0}\)

Step 4: Apply the exponent property for any base to the power of 0

According to the exponent property, any non-zero base raised to the power of 0 is equal to 1: 1 Thus, the simplified expression is: 1

Key Concepts

Exponent Multiplication Property

The exponent multiplication property is an important rule when dealing with powers or exponents in mathematics. This property makes our calculations much easier. It states that if you have the same base being multiplied together with different exponents, you just need to add the exponents.
For example, if you have an expression like \(a^m \times a^n\), you can simplify it to \(a^{m+n}\).

This is because multiplication is essentially repeated addition, and this rule helps consolidate those repeated additions under the law of exponents.

• Keep in mind: this property only applies to terms with the same base.
• When applying, ensure you accurately add the exponents.
• This rule helps in reducing long multiplication processes into simpler calculations.

With the original exercise, we had \(m^{-5} \times m^5\). Applying the property, we just added the exponents: \(-5 + 5 = 0\), simplifying it to \(m^0\).

Simplifying Exponents

Simplifying exponents is essentially about making an expression easier to work with or evaluate. This involves applying certain exponent rules to combine or reduce expressions to their simplest forms. One of the prime strategies is to use the properties of exponents, like the one we just talked about with multiplication of exponents.

When simplifying, you're often looking to:

• Combine like terms using applicable exponent rules.
• Break down complex expressions into more manageable parts.
• Write expressions in a way that reveals the simplest form or answer.

In the given problem, after applying the exponent multiplication property and combining \(-5 + 5\), we simplified to \(m^0\). This is a much neater form, and perfect for further simplification using other rules like the zero exponent rule.

Zero Exponent Rule

The zero exponent rule is one of the most fascinating and useful laws of exponents. It states that any non-zero base raised to the power of zero equals one. Mathematically, it’s expressed as \(a^0 = 1\) where \(a\) is not equal to zero.
This rule comes from the understanding of how exponents build on each other. As you decrease the exponent by one, you are essentially dividing by the base due to the nature of exponent rules, leading to this unique result when the exponent reaches zero.

• This rule only applies to non-zero bases; zero raised to the power of zero is a special case often left undefined.
• It reinforces the idea that simplifying expressions can yield surprising results.

In the exercise at hand, we had \(m^0\). Using the zero exponent rule, this simplifies to 1 since \(m\) is a non-zero base. Hence, the entire expression simplifies neatly to 1. This rule provides a satisfying closure that resolves complex problems seamlessly.