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Chapter 6: Problem 506

Simplify. (a) \(\frac{1}{q^{-10}}\) (b) \(\frac{1}{10^{-3}}\)

Short Answer

Expert verified
(a) \(q^{10}\) (b) \(10^{3}\)

Step by step solution

01

Understand Negative Exponents

Recall that a negative exponent means taking the reciprocal of the base. For example, if you have a term with a negative exponent like \(a^{-b}\), it becomes \(\frac{1}{a^{b}}\).
02

Simplify Part (a)

Given the term \(\frac{1}{q^{-10}}\), we apply the rule for negative exponents. The term becomes \(q^{10}\).
03

Simplify Part (b)

Given the term \(\frac{1}{10^{-3}}\), we again apply the rule for negative exponents. The term becomes \(10^{3}\).

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Key Concepts

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

Reciprocal of Base
When you encounter a negative exponent, it signals that you should take the reciprocal of the base. For instance, consider the expression \(a^{-b}\). The negative exponent \-b\ tells you to flip the base \a\ into its reciprocal. So, \(a^{-b}\) becomes \(\frac{1}{a^{b}}\). This fundamental concept is crucial for simplifying expressions with negative exponents.

In practice this means:
  • All terms with a negative exponent can be re-written.
  • The exponent will become positive after taking the reciprocal of the base.
Always look for negative exponents and immediately think 'reciprocal'. This will make simplifying expressions much easier.
Simplifying Exponents
Simplifying exponents involves reducing expressions to their simplest form. For negative exponents, this often means using the reciprocal of the base. As we saw in the original exercise, \( \frac{1}{q^{-10}} \) simplifies to \( q^{10} \), and \( \frac{1}{10^{-3}}\) turns into \( 10^{3} \).

Here鈥檚 a step-by-step way to simplify exponents:
  • Identify any terms with negative exponents.
  • Rewrite these terms by taking the reciprocal of the base.
  • Change the exponent from negative to positive.
This will help strip down complex expressions into simpler, more manageable forms.
Algebraic Rules
Understanding some basic algebraic rules can make working with exponents easier. Here are a couple of key rules:

  • Product of Powers: When multiplying like bases, add the exponents: \( a^{m} \times a^{n} = a^{m+n} \).
  • Power of a Power: When raising a power to a power, multiply the exponents: \( (a^{m})^{n} = a^{m \times n} \).

Practice Problem: Simplify \( (x^{-3})^{-2} \).

  • Step 1: Apply the power of a power rule, \( (x^{-3})^{-2} \) becomes \( x^{-3 \times -2} \).
  • Step 2: Simplify the exponent: \( x^{6} \).

Knowing these basic rules helps greatly, especially when dealing with negative exponents. Breaking the work into small steps can simplify complex algebra tasks.

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Most popular questions from this chapter

Simplify. (a) \((-7)^{-2}\) (b) \(-7^{-2}\) (c) \(\left(-\frac{1}{7}\right)^{-2}\) (d) \(-\left(\frac{1}{7}\right)^{-2}\)

When Drake simplified \(-3^{0}\) and \((-3)^{0}\) he got the same answer. Explain how using the Order of Operations correctly gives different answers.

Simplify. (a) \(\left(\frac{3}{10}\right)^{-2}\) (b) \(\left(-\frac{2}{c d}\right)^{-3}\)

Divide the monomials. $$\frac{\left(-18 p^{4} q^{7}\right)\left(-6 p^{3} q^{8}\right)}{-36 p^{12} q^{10}}$$

Simplify. (a) \(3 \cdot 4^{-2}\) (b) \((3 \cdot 4)^{-2}\)
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