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Chapter 1: Problem 28

Simplify each exponential expression (leave only positive exponents). $$\frac{5^{3 x+1}}{25}$$

Short Answer

Expert verified
The simplified expression is \(5^{3x-1}\).

Step by step solution

01

Rewrite the Denominator

First, recognize that 25 can be written as an exponential expression with the same base as the numerator. Since 25 equals \(5^2\), rewrite the denominator: \(25 = 5^2\).
02

Apply Exponential Laws

Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the expression. Replace the denominator with its equivalent exponential form: \(\frac{5^{3x+1}}{5^2} = 5^{(3x+1)-2}\).
03

Simplify the Exponent

Simplify the exponent by subtracting the exponents in the numerator and denominator: \(5^{3x+1-2} = 5^{3x-1}\).

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Key Concepts

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

Understanding Positive Exponents
An exponent represents the number of times a base is multiplied by itself. When we talk about positive exponents, we're dealing with standard, non-negative powers that are easy to visualize and work with. Suppose we have a base number and a positive exponent; this means the base is multiplied by itself as many times as the exponent indicates. For example, when we write \(5^3\), it means \(5 \times 5 \times 5\), which equals 125.
  • Positive exponents simplify the multiplication of repeated same numbers.
  • They are foundational to understanding more complex exponential expressions.
Most importantly, when simplifying expressions, we convert negative exponents to positive ones to ensure clarity and consistency in our expressions.
Diving into Exponentiation Laws
Exponentiation laws are valuable tools that simplify working with expressions involving exponents. One such law is the division law, which states that when dividing like bases with exponents, you subtract the exponents. Mathematically, this is expressed as \(\frac{a^m}{a^n} = a^{m-n}\).
  • This rule is particularly helpful when you need to simplify complex fractions of exponential expressions.
  • It ensures that expressions are reduced properly to their simplest form.
By understanding these laws, you also gain insights into how multiplication and powers interact. For example, the Multiplication Law \(a^m \times a^n = a^{m+n}\) shows how exponents are added when multiplying the same bases. Recognizing these laws strengthens your ability to manipulate exponential expressions with confidence.
Steps to Simplifying Exponents
Simplifying exponents is all about following a series of logical steps to reduce a given expression to its simplest form. Let's walk through the example \(\frac{5^{3x+1}}{5^2}\):

Step 1: Rewrite similar bases

First, ensure that all bases are similar. In the example, both the numerator and denominator have a base of 5.

Step 2: Apply exponentiation laws

Use the division rule: subtract the exponents, simplifying \(5^{3x+1}\) by dividing it by \(5^2\). This means:
  • Perform the subtraction: \((3x+1) - 2\).

Step 3: Simplify

Combine the terms to simplify the expression: \(5^{3x+1-2} = 5^{3x-1}\). This example illustrates well how following structured steps can help simplify even complex exponential expressions to achieve the desired result with only positive exponents.

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Most popular questions from this chapter

Perform the indicated operation and simplify. $$\frac{2 x+6}{7} \times \frac{1}{x^{2}-9}$$

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Use interval notation to represent the subset of real numbers that is indicated by the inequality. $$-4

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