Problem 40 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{c d}{b^{2}} \div \frac{d^{2}}{b} $$

Step-by-Step Solution

Answer

\( \frac{c}{b d} \)

Step 1: Write the Division as Multiplication

First, convert the division problem into a multiplication problem by multiplying by the reciprocal of the second fraction: \[ \frac{cd}{b^2} \times \frac{b}{d^2} \]

Step 2: Simplify the Expression

Next, simplify the expression by cancelling out common factors. In this case, both the numerator and the denominator have a common factor of \(d\): \[ \frac{cd \times b}{b^2 \times d^2} \] After canceling out one \(d\): \[ \frac{c b}{b^2 d} \]

Step 3: Cancel Out the Variable b

Simplify further by canceling one \(b\) from the numerator and the denominator: \[ \frac{c}{b d} \]

Key Concepts

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form. The simplest form of a fraction is when the numerator and the denominator are as small as possible while still keeping their ratio the same.

Here’s how you can simplify fractions:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For example, in the original exercise, the expression \frac{cd}{b^2} \times \frac{b}{d^2} simplifies step-by-step. In step 2, we identify common factors in the numerator and the denominator. Both the numerator and the denominator have a common factor of \(d\):\[\frac{cd \times b}{b^2 \times d^2}\]Canceling out \(d\), we get \[\frac{c \times b}{b^2 d}\]Finally, by canceling \(b\) from the numerator and the denominator, we reduce it to \[\frac{c}{bd}\].

Remember, simplifying fractions helps make complex problems more manageable and easier to understand.

Algebraic Expressions

An algebraic expression is any mathematical statement that includes numbers, variables, and operators (like addition or multiplication). It’s like a phrase in mathematics without an equals sign.

In our exercise, the algebraic expression is \(\frac{cd \times b}{b^2 d}\). Here are a few key things to remember about algebraic expressions:

• They often involve variables, which represent unknown values (like \(c\), \(d\), and \(b\) in our example).
• You can simplify these expressions by combining like terms or canceling common factors.
• Operations like addition, subtraction, multiplication, and division apply to both numbers and variables in these expressions.

By understanding the structure and properties of algebraic expressions, you can simplify them, solve for unknown values, and make complex algebraic equations easier to manage.

Reciprocal

The reciprocal of a number is simply one divided by that number. For example, the reciprocal of \(5\) is \(\frac{1}{5}\). The reciprocal of a fraction can be found by swapping its numerator and denominator.

In our exercise, we took the reciprocal of the second fraction \(\frac{d^2}{b}\):\[\frac{b}{d^2}\]
This allows us to change the division problem into a multiplication problem. Using the reciprocal helps to simplify complex division operations, making them more straightforward.

Key points about reciprocals:

• The reciprocal of \(a\) is \(\frac{1}{a}\).
• The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
• Multiplying a number by its reciprocal always gives \(1\).

By understanding and using reciprocals, you can simplify many algebraic operations and solve problems more efficiently.