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Chapter 9: Problem 17

Write each expression with positive exponents, then simplify. $$3^{8} \cdot 3^{-6}$$

Short Answer

Expert verified
The expression simplifies to 9.

Step by step solution

01

Apply the Product of Powers Property

Use the product of powers property which states that when multiplying like bases we add the exponents: \[ 3^{8} imes 3^{-6} = 3^{8 + (-6)} = 3^{8 - 6} \]
02

Calculate the Exponent

Subtract the exponents that are combined: \[ 3^{8 - 6} = 3^{2} \]
03

Evaluate the Expression

Now compute the value of the expression with the simplified exponent: \[ 3^{2} = 9 \]

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Key Concepts

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

Product of Powers Property
The Product of Powers Property is a fundamental rule when dealing with exponents. This property helps simplify expressions that have the same base by combining their exponents. When you multiply numbers with the same base, like our example with the base 3:
  • The rule tells us we can add the exponents together: \( a^m \cdot a^n = a^{m+n} \).
  • It is an efficient way to simplify multiplication problems with exponents.
In our original problem, we had \( 3^8 \cdot 3^{-6} \). By using the Product of Powers Property, we simplified it by adding the exponents, resulting in \( 3^{8 + (-6)} \). This property makes handling large numbers or those with negative exponents much easier.
Simplifying Expressions
Simplifying expressions often involves using algebraic rules, like the Product of Powers Property, to make an equation cleaner or easier to work with. In our exercise:
  • Start by applying exponent rules: After using the Product of Powers Property, we have a single exponential expression \( 3^{8-6} \).
  • Continue by calculating the exponent: This means to perform the arithmetic that combines the exponents, resulting in \( 3^2 \).
  • Simplified outcomes are easier: Simplified expressions are typically more approachable and useful for further computation or understanding.
Breaking down complex problems into simpler parts using these steps can help ensure accuracy and clarity, which is critical in both math and real-life applications.
Positive Exponents
In mathematics, positive exponents indicate how many times a base is multiplied by itself. Working with positive exponents can be easier as it doesn't involve dealing with fractions or reciprocal operations. Here are a few points about positive exponents:
  • Positive exponents are straightforward: \( a^2 \) means \( a \) is used as a factor twice, giving \( a \times a \).
  • Easy computations: For example, \( 3^2 \) equals 9 because \( 3 \times 3 \) results in 9.
  • They contrast negative exponents: While negative exponents represent division, positive exponents are simply repeated multiplication.
In our case, the exercise simplifies to \( 3^2 \), a positive exponent, making it easy to evaluate as 9. Understanding positive exponents helps in recognizing values quickly and efficiently.

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