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Chapter 1: Problem 72

Evaluate. $$ 6 a \div 12 a^{3}, \text { for } a=2 $$

Short Answer

Expert verified
\( \frac{1}{8} \)

Step by step solution

01

Substitute the Value of a

Substitute the value of \(a\) which is 2 into the expression \(6 a \div 12 a^{3}\). This gives us: \[ 6 \times 2 \div 12 \times 2^{3} \]
02

Simplify the Expression

First calculate the exponent: \(2^{3} = 8\). So, the expression becomes: \[6 \times 2 \div 12 \times 8\]. Next, simplify within the numerator and the denominator, yielding: \[ 12 \div 96 \]
03

Calculate the Final Answer

Divide 12 by 96 to get: \[ \frac{12}{96} = \frac{1}{8} \]

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Key Concepts

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

Substitution
Substitution is the process of replacing a variable in an algebraic expression with a given value. This is the first step in solving many algebra problems. In our exercise, we were given the value of \(a\) as 2 and needed to substitute it into the expression \(6a \div 12a^3\).
To do this, we replaced every occurrence of \(a\) with 2, turning the expression into: \[6 \times 2 \div 12 \times 2^3\].
By carefully substituting, we simplify the problem and prepare it for evaluation. Always ensure you substitute accurately to avoid mistakes in later steps.
Evaluation
Evaluation involves calculating the numerical value of an expression after substitution. Once we substituted \(a = 2\), our expression turned into: \[6 \times 2 \div 12 \times 2^3\].
Next, we need to compute the value of each part step-by-step. First, evaluate the exponent: \(2^3 = 8\).
This transforms our expression into: \[6 \times 2 \div 12 \times 8\].
By solving these smaller parts, the final calculation becomes clearer.
Simplifying Expressions
Simplifying expressions is the key to making complex problems manageable. After substituting and evaluating the exponent, our expression was: \(6 \times 2 \div 12 \times 8\).
Start by simplifying within the numerator and the denominator. In our case, the numerator is \(6 \times 2 = 12\) and the denominator is \(12 \times 8 = 96\).
Thus, the expression then simplifies to: \[12 \div 96\].
This simpler form makes the final calculation of the numerical value much easier.
Division of Fractions
Division of fractions often requires simplifying the fractions first. In our example, we reached: \[\frac{12}{96}\].
The next step is to simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For \(12\) and \(96\), the GCD is \(12\).
Thus, \(\frac{12}{96} = \frac{12 \div 12}{96 \div 12} = \frac{1}{8}\).
This gives us the final answer of \(\frac{1}{8}\).
Dividing fractions or simplifying ratios involves understanding how to break down the problem to its simplest form.

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