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Chapter 5: Problem 51

Simplify each expression. Write answers using positive exponents.. $$ 5^{-2} $$

Short Answer

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

Step by step solution

01

Identify the Negative Exponent

The expression given is \(5^{-2}\). Notice that the base of the exponent is 5 and the exponent is -2. A negative exponent indicates the reciprocal of the base raised to the equivalent positive exponent.
02

Convert to Positive Exponent

To convert a negative exponent to a positive one, take the reciprocal of the base. The expression \(5^{-2}\) becomes \(\frac{1}{5^2}\). Now, the exponent is positive.
03

Evaluate the Exponent

Calculate the value of \(5^2\). This means multiplying the base, 5, by itself: \(5 \times 5 = 25\).
04

Write the Final Answer

Now that we've evaluated \(5^2\) as 25, the expression simplifies to \(\frac{1}{25}\). This is the expression with a positive exponent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Exponents
Let's dive into understanding what positive exponents are all about. In mathematics, an exponent tells you how many times a number, called the base, is multiplied by itself. When the exponent is positive, it's pretty straightforward.
  • For instance, in the expression \(5^3\), the base 5 is multiplied by itself three times: \(5 \times 5 \times 5\).
  • The result is 125 because you are simply adding one more 5 to each step of the multiplication.
Positive exponents are a way to express repeated multiplication easily and concisely. They're always easier to work with and interpret because they follow direct multiplication rules. A positive exponent shows the growth of the base number as it gets raised and multiplied by itself multiple times.
Reciprocal
Reciprocal is a unique and essential concept in math, especially when dealing with negative exponents. The reciprocal of a number is 1 divided by that number.
  • For example, the reciprocal of 5 is \(\frac{1}{5}\).
  • It turns the base number upside down, so to speak, so that numerator becomes denominator and vice versa.
When you have a negative exponent, like \(5^{-2}\), the key is to take the reciprocal of the base. This transforms \(5^{-2}\) into \(\frac{1}{5^2}\). Hence, using reciprocals allows us to convert negative exponents into more manageable expressions with positive exponents, maintaining the integrity of the mathematical relationships.
Simplifying Expressions
Simplifying expressions involves rewriting them in the simplest form possible. When given an expression like \(5^{-2}\), the steps can transform it into a more basic form that鈥檚 easier to understand and work with.
  • First, identify any negative exponents and use the reciprocal technique to convert them into positive exponents.
  • In this case, \(5^{-2}\) becomes \(\frac{1}{5^2}\).
  • Next, calculate the positive exponent's value: \(5^2 = 25\).
Finally, the expression \(5^{-2}\) is simplified into \(\frac{1}{25}\). It鈥檚 crucial that when simplifying, you ensure all exponents are positive, making the mathematical expression clear and simple. Through this process, not only are operations performed correctly, but the steps taken ensure clarity and correctness in math.

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