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Chapter 7: Problem 50

For exercises 39-82, simplify. $$ \frac{15 a^{2}}{14 d} \div \frac{a}{d} $$

Short Answer

Expert verified
The simplified form is \( \frac{15 a}{14} \).

Step by step solution

01

Rewrite the Division as Multiplication

To simplify the given expression, start by rewriting the division as multiplication of the reciprocal. The given expression is \[ \frac{15 a^{2}}{14 d} \div \frac{a}{d} \] which can be rewritten as \[\frac{15 a^{2}}{14 d} \times \frac{d}{a} \].
02

Multiply the Numerators and Denominators

Next, multiply the numerators and the denominators of the fractions: \[ \frac{15 a^{2} \times d}{14 d \times a} \].
03

Simplify the Expression

Simplify the expression by canceling out common factors in the numerator and the denominator. The common factors are \(d\) and \(a\). After canceling, the expression simplifies to: \[ \frac{15 a}{14} \].

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Key Concepts

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

Division of Fractions
When dividing fractions, you need to understand that division is the same as multiplying by the reciprocal of the fraction.
The reciprocal of a fraction is simply flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
So, when you see a division like \(\frac{15a^{2}}{14d} \div \frac{a}{d}\), it can be rewritten as \(\frac{15a^{2}}{14d} \times \frac{d}{a}\).
This makes it much easier to work with.
Multiplication of Fractions
Multiplying fractions involves two simple steps:
  • Multiply the numerators together

  • Multiply the denominators together
For example, with \(\frac{15a^{2}}{14d} \times \frac{d}{a}\),
multiply the numerators: 15a\textsuperscript{2} \( \times \) d,
and multiply the denominators: 14d \( \times \) a.
This gives us \(\frac{15a^{2} \times d}{14d \times a} \).
Working step by step makes the multiplication clear and simple.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means reducing the expression to its simplest form.
This often involves canceling out common factors in the numerator and denominator.
  • In our example, \(\frac{15a^{2} \times d}{14d \times a} \)

    • we can cancel out 'd' and 'a' because they appear in both the numerator and the denominator.
    After canceling, we get \(\frac{15a}{14}\).
It's important to always look for common factors to make your expressions as simple as possible.
Reciprocal
A reciprocal is essentially flipping a fraction.
It turns the numerator into the denominator and the denominator into the numerator.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
When solving problems that involve division of fractions,
like \(\frac{15a^{2}}{14d} \div \frac{a}{d}\), the division turns into multiplication with the reciprocal.
This transforms the problem into \(\frac{15a^{2}}{14d} \times \frac{d}{a}\),
making the calculation simpler.

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