Problem 55 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(m^{50}\right)^{10} $$

Step-by-Step Solution

Answer

\((m^{50})^{10} = m^{500}\)

Step 1: Review the Power Rule

The power rule for exponents states that when you have a power raised to another power, you can multiply the exponents. Mathematically, this is expressed as \((a^m)^n = a^{m imes n}\).

Step 2: Apply the Power Rule

To simplify \((m^{50})^{10}\), apply the power rule by multiplying the exponents 50 and 10. This results in \(m^{50 \times 10}\).

Step 3: Calculate the Multiplication

Multiply the exponents 50 and 10 to get 500. Thus, \(m^{50 \times 10} = m^{500}\).

Step 4: Write the Simplified Expression

The expression \((m^{50})^{10}\) simplifies to \(m^{500}\) using the power rule for exponents.

Key Concepts

Exponentiation Rules

Exponentiation is a shorthand way to indicate repeated multiplication. When working with powers and exponents, certain rules can greatly simplify calculations. Here are some key exponentiation rules:

• Product of Powers Rule: When multiplying two expressions with the same base, such as \(a^m \times a^n\), add their exponents to get \(a^{m+n}\).
• Power of a Power Rule: When you have an exponent raised to another exponent, \((a^m)^n\), multiply the exponents, resulting in \(a^{m \times n}\).
• Power of a Product Rule: When raising a product to a power, such as \((ab)^n\), distribute the exponent to each base in the product: \(a^n \times b^n\).

These rules allow you to manipulate and simplify expressions involving exponents with greater ease.

Multiplying Exponents

Multiplying exponents can seem tricky, but it is straightforward once you know the rules. When considering multiplication of powers like \(a^m \times a^n\), remember:
- Both exponents adjust the same base.
In such cases, use the product of powers rule to add the exponents: \(a^{m+n}\). However, if you see an expression like \((a^m)^n\), this is where the power of a power rule comes into play.

• Multiply the exponents: \(a^{m \times n}\), resulting in a much simpler expression.

For instance, simplifying \((m^{50})^{10}\) means multiplying 50 and 10, leading to \(m^{500}\). Thus, these exponents multiply to form an entirely new exponent for the single base.

Simplifying Expressions with Exponents

Simplifying expressions with exponents often involves applying the rules of exponentiation to reduce complexity. The aim is to express the entire expression with as few steps as possible:
- Identify which rule to use based on the structure of the expression.
Take \((m^{50})^{10}\) for example. It initially appears complex, but recognizing it as a power raised to another power, we apply the correct rule to simplify.

• By using the power of a power rule, the expression reduces from a nested form to a simpler expression: \(m^{500}\).

This demonstrates how powerful the exponentiation rules can be. With practice, these steps become intuitive, and anyone can turn seemingly complicated expressions into simpler forms with just a little effort.