Question

Simplify \(5^{-2}\). A. \(\frac{1}{25}\) B. -10 C. -25 D. \(\frac{1}{10}\)

Short answer

The simplified form of \(5^{-2}\) is \(\frac{1}{25}\). Therefore, the correct answer is A.

Step-by-step solution

Understand Negative Exponents

A negative exponent indicates that you need to take the reciprocal of the base and change the sign of the exponent to positive. For instance, \(a^{-b} = \frac{1}{a^b}\). In our case, \(5^{-2}\) means we need to find the reciprocal of \(5^2\) and then evaluate it.

Evaluate the Positive Exponent

Next, calculate the positive exponent. \(5^2 = 5 \times 5 = 25\).

Find the Reciprocal

Now, take the reciprocal of the result from Step 2. The reciprocal of 25 is \(\frac{1}{25}\).

Write the Answer

After simplifying the negative exponent, you find that \(5^{-2} = \frac{1}{25}\).

Key concepts

Reciprocal

When dealing with negative exponents, one important idea is the reciprocal. A reciprocal of a number is simply one divided by that number.
For example, the reciprocal of 5 is \(\frac{1}{5}\).
Reciprocals help us understand how negative exponents work.
Instead of multiplying a base by itself a negative number of times, we take the reciprocal.
For instance, in the exercise above, we had to find the reciprocal of 5 squared because we had a negative exponent of -2.
So, \(5^{-2} = \frac{1}{5^2}\).

Exponent Rules

Understanding exponent rules is vital for working with expressions involving exponents.
Here are some key rules to remember:

• Negative Exponent Rule: \(a^{-b} = \frac{1}{a^b}\) - This tells us that a negative exponent means we take the reciprocal of the base raised to the positive exponent.
• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\) - When multiplying two expressions with the same base, add the exponents.
• Quotient of Powers Rule: \(a^m \/ a^n = a^{m-n}\) - When dividing two expressions with the same base, subtract the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{mn}\) - When raising a power to another power, multiply the exponents.
• Power of a Product Rule: \( (ab)^m = a^m \times b^m\) - When raising a product to a power, apply the exponent to each factor.

Simplifying Exponents

Simplifying expressions with exponents involves applying the exponent rules correctly.
Here’s a step-by-step approach using the exercise example \(5^{-2}\):
First, identify the negative exponent and understand its meaning.
With \(5^{-2}\), the negative exponent tells us to take the reciprocal of the base raised to the positive exponent.
So we write: \(5^{-2} = \frac{1}{5^2} \).
Next, calculate the positive exponent: \(5^2 = 25\).
Finally, take the reciprocal of the result: \( \frac{1}{25} \).
Hence, \(5^{-2} = \frac{1}{25}\).
This shows that simplifying exponents, especially negative ones, requires understanding what the negative symbol signifies and then following the steps accurately.