Problem 54 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. $$ 5^{2} r^{-5}\left(r^{6}\right)^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(25r^{13}\).

Step 1: Simplify the Power of a Power

When you have an expression like \((r^6)^3\), apply the power of a power property \((a^m)^n = a^{m imes n}\). Therefore, \((r^6)^3 = r^{6 \times 3} = r^{18}\).

Step 2: Combine Exponents of Base r

The expression now becomes \(5^2 r^{-5} r^{18}\). Use the property of exponents \(a^m a^n = a^{m+n}\) to combine the exponents for base \(r\). This gives \(r^{-5 + 18}\) which simplifies to \(r^{13}\).

Step 3: Simplify the Entire Expression

Finally, combine the simplified base \(r\) and the numerical part \(5^2\). The complete expression is \(5^2 r^{13}\). Simplify \(5^2\) which is 25. So, the expression simplifies to \(25r^{13}\).

Key Concepts

Understanding the Power of a Power Property

When we encounter exponentiation, it's important to understand the different properties that can simplify the process. One key property is the power of a power property. If you have something like \((a^m)^n\), it can be rewritten as \(a^{m \times n}\). This means you're multiplying the exponents together instead of handling them separately.

For example, in the expression \((r^6)^3\), you apply this property by multiplying 6 by 3. This results in \(r^{18}\). Remember that the base (in this case, \(r\)) stays the same; only the exponents are combined through multiplication.

This property is particularly useful when you have nested powers, allowing for faster simplification and better management of complex equations.

Combining Exponents with the Same Base

Combining exponents is a crucial skill in simplifying expressions in exponentiation. When you have terms with the same base, you can add their exponents together using the property \(a^m \cdot a^n = a^{m+n}\). This holds true as long as the base (\(a\)) is the same for all terms.

In solving the problem, after simplifying the power of a power, we dealt with the expression \(5^2 r^{-5} r^{18}\). Here, the base \(r\) appears twice. To combine them, simply add their exponents: \(-5 + 18\), which results in \(r^{13}\).

This method streamlines evaluating expressions, especially when involving multiple variables or more complex expressions. It's a huge time-saver and reduces the potential for errors.

Simplifying Expressions

Simplifying means making things as straightforward as possible. It involves reducing expressions to their simplest form so they are easy to work with. In expression simplification, every part matters, from coefficients to exponents.

In the exercise at hand, once we combined the exponents for \(r\) to get \(r^{13}\), the next step was addressing the numerical coefficient \(5^2\). This is straightforward: \(5^2 = 25\).

Thus, the entire expression transitions from a complex combination of terms into something more manageable: \(25r^{13}\). Simplifying expressions is essential for clearer communication and efficiency in solving mathematical problems. It makes interpreting results easier and ensures accuracy in further analysis or calculations.