Problem 39 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ x^{3} \cdot \frac{4 a b}{x} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(4ab\cdot x^2\).

Step 1: Identify the Numerator and Denominator

In this expression, the numerator is \(4ab\cdot x^3\), and the denominator is \(x\).

Step 2: Cancel out the common factors

Both the numerator and the denominator have a common factor of \(x\). We can cancel out the common factor of \(x\) by dividing both the numerator and the denominator by \(x\): $$ \frac{4ab\cdot x^3}{x} = \frac{4ab\cdot x^3}{1} \cdot \frac{1}{x} = 4ab\cdot x^2 $$

Step 3: Write the final simplified expression

The simplified expression after performing the multiplications and divisions is: $$ x^{3} \cdot \frac{4 a b}{x} = 4ab\cdot x^2 $$

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. This process can make equations more manageable and easier to solve. When simplifying, one should combine constants with constants, coefficients with coefficients, and similar variables with similar variables.

For example, to simplify an expression like \( x^{3} \times \frac{4ab}{x} \), you would look for terms that are alike and that can be combined. In this case, the common terms involve the variable \(x\). By identifying and combining these like terms, you ultimately reduce the complexity of the expression.

Canceling Common Factors

Canceling common factors is a critical step in simplifying algebraic fractions. It involves finding factors that are the same in both the numerator and the denominator and eliminating them, as they effectively divide to one.

To cancel common factors, first, factorize the numerator and the denominator into their prime components, if possible. Then, identify any common factors between them. These common factors can be 'canceled out,' which means they can be divided out of the fraction. For instance, in the expression \( \frac{x^{3} \cdot 4ab}{x} \), the variable \(x\) is present in both the numerator and denominator and can be canceled, simplifying the expression to \(4ab \cdot x^{2}\).

Exponent Rules

Understanding the rules of exponents, also known as 'powers,' is essential when simplifying algebraic expressions that include terms raised to a power. Key rules include:

• The 'Product of Powers' rule, which states that when multiplying two powers with the same base, you can add the exponents: \( x^{m} \times x^{n} = x^{m+n} \).
• The 'Quotient of Powers' rule which indicates that when dividing two powers with the same base, you can subtract the exponents: \( \frac{x^{m}}{x^{n}} = x^{m-n} \).
• The 'Power of a Power' rule expresses that when raising a power to another power, you multiply the exponents: \( (x^{m})^{n} = x^{mn} \).

In the given exercise, we apply the 'Quotient of Powers' rule. Here, \( x^{3} \times \frac{4ab}{x} \), becomes \(4ab \cdot x^{2}\) because we subtract the exponent of \(x\) in the denominator from the exponent of \(x\) in the numerator (\(3 - 1 = 2\)).