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Chapter 10: Problem 55

Simplify. Write the answers with positive exponents only. $$y^{-12} y^{14}$$

Short Answer

Expert verified
The simplified expression is \(y^{2}\).

Step by step solution

01

Apply the Product of Powers Property

When multiplying expressions that have the same base, add their exponents. For the given expression \(y^{-12} y^{14}\), add the exponents: \(-12 + 14\).
02

Simplify the Exponents

Calculate the sum of the exponents: \(-12 + 14 = 2\).
03

Rewrite with Positive Exponents

The resulting expression is \(y^{2}\). Since the exponent is already positive, no further steps are needed.

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Key Concepts

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

Product of Powers Property
When you multiply two expressions with the same base, you use the product of powers property. This rule says that you simply add the exponents together. For example, in the expression\( y^{-12} y^{14}\), you keep the base \( y \) and add the exponents. This helps in simplifying complex expressions easily.
Positive Exponents
When working with exponents, it's important to express your final answer with positive exponents. Positive exponents indicate how many times you multiply the base by itself. In the case of our expression, once we've added the exponents using the product of powers property, we get \( y^{2} \). Since 2 is already positive, there's no need for further conversion. Remember, positive exponents make the results easier to read and less confusing.
Simplification Steps
Simplifying expressions with exponents involves a few clear steps:

  • First, identify the base and make sure they are the same.
  • Next, use the product of powers property to add the exponents. In our example, we calculated -12 + 14 to get 2.
  • Finally, verify that your exponents are positive and adjust if necessary. Here, \( y^{2} \) is already positive, so it's the simplified answer.

These steps help ensure you simplify complex expressions accurately and efficiently.

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