Problem 99 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{2 x^{3} y^{2} z}{x y z} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^2y \).

Step 1: Write the Expression

The expression given is \( \frac{2x^3 y^2 z}{x y z} \). This is a fraction with a numerator and a denominator.

Step 2: Simplify the Coefficients

The coefficient in the numerator is 2. There are no numerical coefficients in the denominator, so the coefficient remains 2.

Step 3: Simplify the x Terms

Look at the powers of \( x \) in both the numerator and the denominator. In the numerator, \( x^3 \), and in the denominator, \( x^1 \). Using the laws of exponents, divide the powers: \( x^{3-1} = x^2 \).

Step 4: Simplify the y Terms

Look at the powers of \( y \) in both the numerator and the denominator. In the numerator, \( y^2 \), and in the denominator, \( y^1 \). Using the laws of exponents, divide the powers: \( y^{2-1} = y^1 \).

Step 5: Simplify the z Terms

Look at the powers of \( z \) in both the numerator and the denominator. In the numerator, \( z^1 \), and in the denominator, \( z^1 \). Using the laws of exponents, divide the powers: \( z^{1-1} = z^0 = 1 \).

Step 6: Combine All Simplified Parts

Put together all the simplified components of the expression to get the final simplified form: \( 2x^2y \).

Key Concepts

Exponents

Exponents are a foundational concept in mathematics where a number, called the base, is multiplied by itself a certain number of times, indicated by the exponent. For example, in the expression \( x^3 \), the base is \( x \) and it is raised to the power of 3, meaning \( x \) multiplied by itself three times: \( x \times x \times x \).
Understanding exponents makes simplifying expressions more manageable, especially when the same base appears in both the numerator and denominator of a fraction. This property allows you to simplify expressions by using the rules of exponents.

• Positive whole number exponents represent standard multiplication repeated several times.
• Zero exponents: Any non-zero number raised to the power of zero equals one, such as \( z^0 = 1 \).

When working with exponents, paying attention to the base and exponent is crucial. Missing or combining incorrectly might lead to confusion or incorrect results. Remember, simplifying with exponents is akin to finding the most efficient way to express multiplication!

Numerator and Denominator

In any fraction, the numerator and denominator hold vital roles. The numerator is the top part of the fraction, while the denominator is at the bottom. Understanding each component helps in the process of simplifying expressions.
For example, consider the fraction \( \frac{2x^3 y^2 z}{x y z} \). Here, \( 2x^3 y^2 z \) is the numerator, and \( x y z \) is the denominator. Simplification involves reducing these components where possible by following mathematical rules and understanding component behaviors.

• Numerators represent the portion of each whole part.
• Denominators express into how many parts the whole is divided.
• Simplification seeks common factors between the numerator and denominator.
• Divide both components by their common factors to reach simplest form.

Recognizing how exponents within both the numerator and denominator interact is key to achieving simplification, as seen through applying the laws of exponents to divide and reduce the entire expression.

Laws of Exponents

The laws of exponents are essential when simplifying expressions involving exponents. These laws help us understand how to manipulate numbers more efficiently. Key laws include:

• Product of powers law: \( x^a \times x^b = x^{a+b} \)
• Quotient of powers law: \( \frac{x^a}{x^b} = x^{a-b} \), used in the example to reduce \( x^3 \) by \( x^1 \) to get \( x^2 \).
• Power of a power law: \( (x^a)^b = x^{a \times b} \)

Using these laws, we can break down complex expressions step-by-step. They help identify which powers to subtract when similar bases appear in both the numerator and denominator, as demonstrated in reducing \( y^2 \) by \( y^1 \) to form \( y^1 \).
These laws not only simplify the current expression but serve as fundamental tools in broader mathematical applications as well. Understanding and practicing these applications will make manipulating and solving mathematical problems much easier.