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Chapter 5: Problem 76

Simplify. Do not use negative exponents in the answer. \(\frac{b^{-2} b^{-3}}{b^{-9}}\)

Short Answer

Expert verified
The simplified expression is \(b^4\).

Step by step solution

01

Apply Exponent Rules

When multiplying terms with the same base, you add the exponents. Therefore, start by combining the exponents in the numerator. Given: \[ \frac{b^{-2} b^{-3}}{b^{-9}}\] Combine the exponents in the numerator: \[b^{-2 + (-3)} = b^{-5}\] Thus, the expression becomes: \[\frac{b^{-5}}{b^{-9}}\]
02

Use the Quotient Rule for Exponents

When dividing terms with the same base, subtract the exponents of the denominator from the exponents of the numerator. Use this rule to simplify the expression: \[\frac{b^{-5}}{b^{-9}} = b^{-5 - (-9)} \]Subtracting gives: \[b^{-5 + 9} = b^{4}\]
03

Provide the Final Answer Without Negative Exponents

The simplified expression with no negative exponents is \[b^4\].

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Key Concepts

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

Quotient Rule for Exponents
The Quotient Rule for Exponents is a powerful tool when simplifying algebraic expressions, especially those involving variables with exponents. If you come across a fraction where both the numerator and the denominator have the same base but different exponents, this rule comes into play. Remember this simple idea:
  • When dividing two expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.
  • The mathematical expression for this is: \( \frac{a^m}{a^n} = a^{m-n} \).
To apply this rule, ensure that the bases are identical. In our example, both the numerator and the denominator have the base \( b \). We used the Quotient Rule to transform \( \frac{b^{-5}}{b^{-9}} \) to \( b^{-5 - (-9)} \). Don't forget to subtract those exponents carefully; notice how subtracting a negative is like adding!
Negative Exponents
Negative exponents can be confusing initially, but they aren't as complicated as they may seem. A negative exponent signifies the inverse of that number with a positive exponent. Consider the following:
  • If you have a negative exponent, such as \( b^{-n} \), it equals \( \frac{1}{b^n} \).
  • This rule highlights that \( b^{-n} \) is essentially the reciprocal of \( b^n \).
In our exercise, the term \( b^{-2} \cdot b^{-3} \) simplifies to \( b^{-5} \). Initially, this may seem peculiar, but when you think of \( b^{-5} \) as \( \frac{1}{b^5} \), it becomes easier to understand. When dealing with negative exponents, always aim to rewrite them positively in your final answers. This makes expressions clearer and aligns with conventional mathematical representations.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is about making them as straightforward as possible. At times, this process involves multiple steps, such as combining like terms and ensuring there are no negative exponents in your final answer.
  • Start by applying any relevant rules (such as exponent rules) to combine like items.
  • In our exercise, \( b^{-2} \cdot b^{-3} = b^{-5} \), settling all same-base terms in the numerator.
  • Next, use the Quotient Rule to simplify further, turning \( \frac{b^{-5}}{b^{-9}} \) into \( b^4 \).
  • Finally, ensure no negative exponents are left in your result, which makes \( b^4 \) the simplest form.
By focusing on these steps, you can tackle even the most complicated algebraic expressions. Your goal is always to achieve the simplest and most readable form of the expression without altering its value.

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