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Chapter 7: Problem 24

Simplify by removing a factor equal to 1. $$ \frac{7 a^{3}}{21 a} $$

Short Answer

Expert verified
\( \frac{a^{2}}{3} \)

Step by step solution

01

Identify the common factors in the numerator and the denominator

Look at the expression \(\frac{7 a^{3}}{21 a}\). Identify any common factors in the numerator and the denominator. The common factor here is 7 and a.
02

Factor out the common terms

Factor out the common terms from the numerator and denominator. The expression becomes: \(\frac{7 \times a^{3}}{21 \times a}\). 21 can be further factored as \(7 \times 3\). Therefore, the expression simplifies to \(\frac{7 \times a^{3}}{7 \times 3 \times a}\).
03

Cancel the common factors

Cancel the common factors from the numerator and denominator. Both the numerator and denominator contain the factors 7 and a: \( \frac{ot{7} \times a^{3}}{ot{7} \times 3 \times a}\) which simplifies to \(\frac{a^{3}}{3a}\).
04

Simplify the remaining terms

Now, simplify the remaining terms. \( \frac{a^{3}}{3a} = \frac{a^{3-1}}{3} = \frac{a^{2}}{3} \). This is because when dividing terms with the same base, subtract their exponents.

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Key Concepts

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

Understanding Algebraic Fractions
Algebraic fractions involve ratios of algebraic expressions instead of simple numbers. An example is \(\frac{7 a^{3}}{21 a}\), which has polynomials in both its numerator and denominator.
Understanding algebraic fractions is key in simplifying complex expressions.
Always look for patterns or common factors within these fractions.
The Role of Common Factors
Common factors are numbers or variables that can divide both the numerator and the denominator evenly.
In our expression \(\frac{7 a^{3}}{21 a}\), 7 and 'a' are common factors.
Identifying and factoring out these common terms simplify the expression, making it easier to handle.
For instance, by factoring out 7 and 'a', we reach \(\frac{7 \times a^{3}}{7 \times 3 \times a}\).
This step is crucial to further reduce the fraction.
Exponent Rules Simplified
Exponent rules help manage and simplify expressions involving powers.
One important rule is when dividing two terms with the same base: subtract the exponents.
For instance, \(\frac{a^{3}}{a}\) simplifies to \(\frac{a^{3-1}}{1} = a^{2}\).
Always keep this in mind for accurate simplifications in algebraic fractions.
Technique of Canceling Terms
Canceling terms involves removing common factors from both the numerator and the denominator.
For the expression \(\frac{7 \times a^{3}}{7 \times 3 \times a}\), the '7' and 'a' terms cancel out, leaving \(\frac{a^{2}}{3}\).
Ensuring the numerator and denominator are reduced as much as possible is key in simplifying fractions.
This technique not only simplifies but also makes further computations easier.

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