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Chapter 8: Problem 71

Perform the operations and simplify. Leave answers in exponential form. \(57 / 351 / 3\)

Short Answer

Expert verified
\((3^3 \times 13)^{-1}\) or \(3^{-3} \times 13^{-1}\)

Step by step solution

01

Simplify the Division from Left to Right

First, perform the division from left to right as this is the correct order of operations for a series of divisions. Start with dividing 57 by 351: \[\frac{57}{351} = \frac{57}{3 imes 117} = \frac{1}{3 \times 39} = \frac{1}{3 \times 3 \times 13} = \frac{1}{9 \times 13}\]
02

Complete the Division

Now divide the result \(\frac{1}{9 \times 13}\) by 3:\[\frac{1}{9 \times 13 \times 3} = \frac{1}{27 \times 13} = \frac{1}{351}\]
03

Write the Answer in Exponential Form

Convert the result into exponential form. Since 351 can be factored into prime factors as \(3^3 \times 13\), \[\frac{1}{351} = 351^{-1} = (3^3 \times 13)^{-1} = 3^{-3} \times 13^{-1}\] The result still represents the division in exponential form.

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Key Concepts

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

Order of Operations
In mathematics, performing operations in the correct order is crucial to getting accurate results. This rule is often remembered by the acronym PEMDAS, which stands for:
  • P: Parentheses
  • E: Exponents
  • M/D: Multiplication/Division
  • A/S: Addition/Subtraction
The rule for Division and Multiplication is that they should be carried out from left to right as they appear in the equation.
In this exercise, we have the expression \(57 / 351 / 3\). We perform the operations from left to right as follows:
First, divide 57 by 351. Then, take the result and divide it by 3. Ignoring the proper sequence would lead to incorrect results. It鈥檚 essential to respect this order to maintain mathematical accuracy.
Division Simplification
Simplifying division involves breaking down complex expressions into their simplest form. This can often include reducing fractions by common factors.
First, simplify \(\frac{57}{351}\). We simplify by finding and canceling common factors.
  • Factor 351: It's equal to \(3 \times 3 \times 39\) or \(3^2 \times 13\).
  • Cancel the common factor of 3 from both the numerator and the denominator.
This simplifies to \(\frac{1}{9 \times 13}\).
We then take this simplified version and divide it by 3. When we divide by 3, the denominator becomes \(9 \times 13 \times 3\), which simplifies further to \(27 \times 13\). Thus, our simplified division is \(\frac{1}{351}\).
Prime Factorization
Prime factorization is expressing a number as a product of its prime numbers. This technique simplifies operations like division, especially when dealing with large numbers.
In our example, 351 is broken down using its prime factors. Prime factors are prime numbers that multiply to give the original number.
  • 351 can be expressed as \(3^3 \times 13\).
To write \(\frac{1}{351}\) in exponential form, we use these prime factors.
We represent the division in exponential terms as \(3^{-3} \times 13^{-1}\).
This shows the power and simplicity of exponential form for expressing division with prime factors, keeping operations neat and understandable.

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