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Chapter 10: Problem 14

Simplify. $$\frac{9^{4}}{9^{3}}$$

Short Answer

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Step by step solution

01

Understand the Problem

Identify that the problem is asking to simplify the fraction involving exponents.
02

Recall the Laws of Exponents

Use the law of exponents which states \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents.
03

Apply the Law of Exponents

Apply the law to the given expression: \(\frac{9^4}{9^3} = 9^{4-3}\).
04

Simplify the Exponent

Subtract the exponents: \(4 - 3 = 1\). Thus, the expression simplifies to: \(9^1 = 9\).

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Key Concepts

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

Laws of Exponents
To simplify fractions with exponents, you need to understand the basic laws of exponents. The key law used in the given problem is:
  • The Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

This rule can be applied when you have the same base, and you are dividing two exponents. In this case, you simply subtract the exponent in the denominator from the exponent in the numerator.

Other common laws of exponents to keep in mind include:
  • The Product Rule: \(a^m \cdot a^n = a^{m+n}\)
  • The Power Rule: \((a^m)^n = a^{mn}\)
  • Zero Exponent Rule: \(a^0 = 1\)
  • Negative Exponent Rule: \(a^{-m} = \frac{1}{a^m}\)

Understanding these rules can give you a robust toolkit for simplifying algebraic expressions involving exponents.
Exponent Subtraction
In the problem \(\frac{9^4}{9^3}\), the key step involves subtracting exponents. When dividing like bases, you can simplify the expression by subtracting the exponent in the denominator from the exponent in the numerator.

Here's a step-by-step breakdown:
  • Recognize that both the numerator and the denominator have the same base, which is 9.
  • Apply the quotient rule (from the Laws of Exponents): \(\frac{9^4}{9^3} = 9^{4-3}\)
  • Calculate the difference: \4 - 3 = 1\
  • Simplify the expression: \9^{1}\

Therefore, \9^{1} = 9\. By using exponent subtraction, the fraction is simplified into a much simpler expression.
Simplifying Algebraic Expressions
Simplifying algebraic expressions often involves combining like terms and using the laws of exponents. Here's what you should keep in mind:
  • Identify like terms: Like terms have identical variables raised to the same power.
  • Use exponent rules: Applying rules like the product and quotient rule helps in reducing complex terms.
  • Combine terms: Add or subtract coefficients of like terms to simplify the expression.

In the example of \(\frac{9^4}{9^3}\), we applied the quotient rule to simplify it directly. In more complex expressions, the same principles apply to each term.

By mastering these basics, you can handle both simple and complex algebraic expressions more effectively.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{-2}$$

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Simplify. Write the answers with positive exponents only. $$\left(4 m^{3} n^{-5}\right)\left(m^{-4} n^{6}\right)$$

Simplify the expression. $$\left(2 p^{2}\right)^{3}(p q)^{2}$$

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