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Chapter 5: Problem 23

Express using positive exponents and, if possible, simplify. $$3 a^{8} b^{-6}$$

Short Answer

Expert verified
\frac{3 a^{8}}{b^6}

Step by step solution

01

Identify Negative Exponents

Look at the given expression to find any terms with negative exponents. The term with a negative exponent here is $$b^{-6}$$.
02

Rewrite Negative Exponents as Positive

Apply the rule $$a^{-n} = \frac{1}{a^n}$$ to rewrite negative exponents as positive. The term $$b^{-6}$$ becomes $$\frac{1}{b^6}.$$ So, the expression now looks like: $$3 a^{8} \cdot \frac{1}{b^6}.$$
03

Simplify the Expression

Combine the terms to simplify the expression. This gives: $$\frac{3 a^{8}}{b^6}.$$

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Key Concepts

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

positive exponents
Positive exponents are the usual way we see exponents when they are not negative or zero. They tell us how many times to multiply the base by itself. For example, in the term \(a^8\), the exponent 8 tells us to multiply 'a' by itself 8 times.
Here's a quick recap on how positive exponents work:
  • If you see an expression like \(x^3\), it means you multiply 'x' three times: \(x \times x \times x\).
  • Exponents apply only to the number or variable directly in front of them unless parentheses are used.
  • Positive exponents make numbers grow larger as they represent repeated multiplication.
In our exercise, the term \(a^8\) is already in its positive exponent form, meaning we don't need to change it.
simplifying expressions
Simplifying expressions is the process of making them easier to work with by combining like terms or reducing fractions. In the context of algebraic expressions, it often involves rearranging the expression to its most concise form.
Let's break down the steps from our example:
  • First, identify any negative exponents, like \(b^{-6}\).
  • Rewrite every term with a negative exponent to have a positive exponent. For example, \(b^{-6}\) becomes \(\frac{1}{b^6}\).
  • Combine the terms. Look at the fractions and see if they can be combined into one term. In the exercise, \(3 a^8 \times \frac{1}{b^6}\) simplifies to \(\frac{3 a^8}{b^6}\).
Simplifying helps you understand the expression better and makes it easier to solve, especially when it appears in larger equations or problems.
negative exponents
Negative exponents can be tricky, but they follow specific rules that make them manageable. Essentially, a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.
Here's how to handle negative exponents:
  • Use the rule \(a^{-n} = \frac{1}{a^n}\). This means instead of multiplying, you divide.
  • For example, \(b^{-6}\) turns into \(\frac{1}{b^6}\). This flips the base to the denominator while making the exponent positive.
  • Apply this transformation to each term individually before combining them.
In our provided exercise, applying this rule to \(b^{-6}\) helped us rewrite the original expression as \(3 a^8 \times \frac{1}{b^6}\). By then combining the terms, we got the simplified fraction: \(\frac{3 a^8}{b^6}\).

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