Question

Divide and simplify. $$44 a^{2} b^{3} c^{4} \div 11 a^{2} b c$$

Short answer

The simplified expression is 4 b^{2} c^{3}.

Step-by-step solution

Write the division as a fraction

Begin by expressing the division of the two algebraic expressions as a fraction. In this case, place the dividend (44 a^{2} b^{3} c^{4}) as the numerator and the divisor (11 a^{2} b c) as the denominator.

Cancel common factors

Simplify the fraction by canceling out any common factors from the numerator and the denominator. Since 44 and 11 are both divisible by 11, 11 cancels out from these numbers, and you are also able to cancel out a^{2} from both the numerator and the denominator since they are common to both.

Simplify the remaining expression

After canceling the common factors, you are left with b^{3} in the numerator and b in the denominator which simplifies to b^{2}, and c^{4} in the numerator and c in the denominator which simplifies to c^{3}. No further simplification can be done with the variables as there are no more common factors.

Key concepts

Simplify Algebraic Expressions

To simplify algebraic expressions is to rewrite them in their simplest form, which involves reducing them to the fewest terms possible without changing their value. The goal is to make the expressions easier to understand and work with. Simplifying can involve combining like terms, which are terms that have the same variables and exponents, and performing operations with numbers and variables.

For example, consider the expression 3x + 2x. We can simplify this by combining like terms to get 5x. Simplifying expressions makes subsequent algebraic operations, such as addition, subtraction, multiplication, and division, more straightforward. Simplification is not just about making an expression shorter; it is about making the expression as clear and easy to work with as possible.

Division of Algebraic Terms

The division of algebraic terms involves dividing one algebraic expression by another. To perform division, you can write algebraic terms in the form of a fraction where the term being divided is the numerator and the term by which you are dividing is the denominator. The division operation is complete when all like terms are simplified, and any common factors are canceled.

For instance, in the expression \(\frac{x^2 - 9}{x - 3}\), we can factor the numerator to \((x + 3)(x - 3)\) and then divide by the denominator \(x - 3\), simplifying the expression to \(x + 3\). This process can be used with numerical coefficients as well as variable terms, but it's important to ensure that the division is valid and that you’re not dividing by zero, as this is undefined.

Canceling Common Factors

Canceling common factors in algebraic fractions is a powerful technique used to simplify expressions. It involves dividing both the numerator and the denominator by the same non-zero factor, thus reducing the expression to its simplest form. To cancel common factors, both the numerator and the denominator must be factored into their prime factors or into any common terms they share.

The common factors can be variables, numbers, or algebraic terms. Remember, only like terms (terms that have the exact same variable part) can cancel out. For example, in the expression \(\frac{6x^2y}{3xy}\), both the numerator and the denominator share the factor \(3xy\). Canceling it out simplifies the expression to \(2x\).

Careful attention to the exponent values

When dealing with variables, the exponent values play a crucial role. Only the lowest powers of common variable factors can be canceled out. In a situation where we have \(\frac{a^m}{a^n}\), if \(m>n\), we subtract the exponents and leave the result in the numerator: \(a^{m-n}\). Conversely, if \(m