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Chapter 9: Problem 38

Divide as indicated. Write your answer using only positive exponents. $$10^{-3} \div 10$$

Short Answer

Expert verified
The answer is \(\frac{1}{10^4}\).

Step by step solution

01

Apply the Division Rule for Exponents

When dividing powers with the same base, subtract the exponent of the divisor from the exponent of the dividend. The rule is: \(a^m \div a^n = a^{m-n}\). In this problem, both terms have base 10, so subtract the exponents: \(-3 - 1\).
02

Simplify the Exponent

Perform the subtraction from Step 1: \(-3 - 1 = -4\). Therefore, the expression simplifies to: \[10^{-4}\].
03

Convert Negative Exponent to Positive Exponent

To write the answer with positive exponents, recall that \(a^{-n} = \frac{1}{a^n}\). Thus, \(10^{-4}\) can be rewritten as \(\frac{1}{10^4}\).

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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 straightforward to understand. They signify repeated multiplication of the base. For example, when you see a positive exponent like \(10^3\), it means you multiply the base, 10, by itself three times: \(10 \times 10 \times 10\). This results in 1000. Positive exponents indicate how many times the base is used in multiplication.
When working with positive exponents, remember:
  • They make calculations larger unless the base is a fraction.
  • They are simple to interpret: just multiply the base as many times as the exponent indicates.
  • In equations, they provide straightforward operations easy to solve and understand.
Negative Exponents
Negative exponents might seem tricky at first, but they're quite manageable. A negative exponent indicates division rather than multiplication. Specifically, it's the reciprocal of the base raised to the opposite positive exponent.
For example, \(10^{-3}\) is the same as \(\frac{1}{10^3}\). This is vital when simplifying expressions since it helps convert negative exponents into positive ones, which are easier to handle and visualize.
  • Negative exponents indicate division of 1 by the base raised to the corresponding positive exponent.
  • They provide a way to express very small numbers conveniently, as opposed to very large numbers expressed by positive exponents.
  • They allow expressions to be rewritten with only positive exponents, providing a consistent approach to simplification.
Division Rule for Exponents
The division rule for exponents simplifies expressions with shared bases. When dividing like bases, you subtract the exponent of the divisor from the exponent of the dividend. For instance, in the expression \(a^m \div a^n\), apply: \(a^{m-n}\).
In the given problem \(10^{-3} \div 10^1\), both terms have the base of 10. Thus, simply subtract: \(-3 - 1 = -4\). It results in \(10^{-4}\), and further steps convert this into \(\frac{1}{10^4}\), offering a result solely with positive exponents.
Remember these important points:
  • The bases must be the same to apply this rule.
  • The operation involves straightforward subtraction of exponents, simplifying the expression efficiently.
  • This rule helps in maintaining a base number while adjusting the power, making it easier to handle exponent division problems.

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Most popular questions from this chapter

Expand and multiply. $$(x-4)^{2}$$

Simplify the following expressions. \(\left(9 x^{4}\right)^{2}\)

Simplify the following expressions. \(\left(a^{5} b^{3}\right)^{2}\left(a b^{3}\right)^{4}\left(a^{3} b\right)^{5}\)

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3 \times 10^{7}\right)\left(8 \times 10^{4}\right)}{6 \times 10^{5}}$$

Multiply the following expressions. \(\left(7 a^{3} b^{2}\right)\left(8 a b^{3}\right)\)
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