Problem 10 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$(-5)^{-2}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{25}\).

Step 1: Understand Negative Exponents

Recall that any number raised to a negative exponent is equal to the reciprocal of that number raised to the opposite positive exponent. For a base \( a \) and exponent \( -n \), it can be expressed as: a^{-n} = \frac{1}{a^n}.

Step 2: Apply the Reciprocal Rule

Apply the reciprocal rule to \((-5)^{-2}\). This means you need to take the reciprocal of \(-5\) and then raise it to the power of \(2\). \((-5)^{-2} = \frac{1}{(-5)^2}\).

Step 3: Calculate the Positive Exponent

Now, calculate \((-5)^2\). When you square a negative number, the result is positive, so:\((-5)^2 = 25\).

Step 4: Simplify the Expression

Substitute the result of Step 3 back into the expression from Step 2. So:\(\frac{1}{(-5)^2} = \frac{1}{25}\).

Step 5: Conclusion

You have simplified the expression. The final answer is \(\frac{1}{25}\).

Key Concepts

reciprocal rule

When working with negative exponents, understanding the reciprocal rule is essential. A negative exponent signifies the reciprocal of the base raised to the opposite positive power. For example, converting a negative exponent involves flipping the fraction. If the base is 'a' and the exponent is '-n', it becomes the reciprocal: \( a^{-n} = \frac{1}{a^n} \).

This rule helps transform complex expressions into simpler forms. To apply this rule, always remember:

• Take the reciprocal of the base.
• Flip the sign of the exponent to positive.

By mastering the reciprocal rule, you can tackle negative exponent problems with ease, converting them into expressions that are easier to compute and understand.

positive exponent

Making sense of positive exponents is simplified when you replace negative exponents through the reciprocal rule. Once you have the base as a positive exponent: \((-5)^{-2} = \frac{1}{(-5)^2}\), your task becomes straightforward.

Calculating a power with a positive exponent involves multiplying the base by itself as many times as the exponent indicates. For instance, \((-5)^2\) means multiplying \(-5\) by itself, resulting in \(25\).

Here are some points to remember about positive exponents:

• Positive exponents indicate repeated multiplication.
• A negative base raised to an even exponent makes the result positive.

Recognizing these properties allows simplification of expressions to become more direct, making complex problems more approachable.

simplifying expressions

Simplifying expressions involves breaking down and organizing components to arrive at a straightforward form. With exponents, this often requires applying the reciprocal rule and solving for positive exponents.

Take the expression \((-5)^{-2}\). By using these steps:

• Convert the negative exponent using the reciprocal rule: \((-5)^{-2} = \frac{1}{(-5)^2}\).
• Solve the positive exponent expression: \((-5)^2 = 25\).
• Substitute and simplify: \(\frac{1}{25}\).

Simplification not only leads to the final answer but also enhances your understanding of mathematical properties and operations, making larger, more complex expressions easier to handle and solve.