Problem 57 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{6 y^{3} z}{2 y z^{2}} $$

Step-by-Step Solution

Answer

\( \frac{3y^{2}}{z} \)

Step 1: Simplifying the Coefficients

The fraction \( \frac{6 y^{3} z}{2 y z^{2}} \) has two coefficients: 6 in the numerator and 2 in the denominator. Simplify the fraction of these coefficients by dividing both by 2.\[ \frac{6}{2} = 3 \]

Step 2: Simplifying the y Terms

The variable 'y' has an exponent of 3 in the numerator and an exponent of 1 in the denominator. To simplify, subtract the exponent in the denominator from the exponent in the numerator. \[ y^{3-1} = y^{2} \]

Step 3: Simplifying the z Terms

The variable 'z' has an exponent of 1 in the numerator and an exponent of 2 in the denominator. Simplify by subtracting the exponents, resulting in: \[ z^{1-2} = z^{-1} \]

Step 4: Eliminate Negative Exponent

Since we have a negative exponent for \( z \), move \( z \) to the denominator to eliminate the negative exponent: \[ z^{-1} = \frac{1}{z} \]

Step 5: Combine Simplified Parts

Combine all the simplified parts together to form the final simplified expression: \[ \frac{3y^{2}}{z} \]

Key Concepts

Negative Exponents

In algebra, negative exponents can sometimes seem confusing, but their purpose is to make expressions more manageable. When a variable is raised to a negative exponent, it indicates that the variable should be moved to the opposite part of the fraction. For example, if you see a negative exponent in the numerator like in our problem with \( z^{-1} \), it tells us to move \( z \) to the denominator. This transforms \( z^{-1} \) into \( \frac{1}{z} \), hence eliminating the negative exponent.

The general rule is: \( a^{-n} = \frac{1}{a^n} \) for any non-zero number \( a \) and positive integer \( n \). Understanding and applying this rule is essential when simplifying algebraic expressions. Not only does it make expressions look nicer, but it also helps in further calculations.

Simplifying Fractions

Simplifying fractions is like streamlining the terms to their most basic form, making them easier to work with. In the expression \( \frac{6 y^{3} z}{2 y z^{2}} \), the task is to simplify both numerical coefficients and the algebraic terms.

Begin with the coefficients: Divide the numerator coefficient (6) by the denominator coefficient (2), resulting in 3. This simplifies the coefficient part of the fraction from \( \frac{6}{2} \) to 3.

Next, handle the algebraic variables. For \( y \), subtract the exponent in the denominator from the one in the numerator: \( y^{3-1} \), which simplifies to \( y^2 \). By following these basic fraction simplification rules, you can manage complex expressions with ease.

• Always reduce numerical coefficients as you would with regular fractions.
• Simplify each variable separately by subtracting exponents.

Algebraic Variables

Algebraic variables represent unknown quantities and can often be manipulated using various algebraic rules. In our initial expression \( \frac{6 y^{3} z}{2 y z^{2}} \), we have variables \( y \) and \( z \) with different exponents.

Each variable should be dealt with separately, focusing on combining like terms and following the laws of exponents. For \( y \), it’s about subtracting exponents between the numerator and the denominator. With \( z \), the original exponent rule led to a negative exponent.

• Recognize and apply exponent rules effectively to each variable.
• A variable with no visible exponent actually means it has an exponent of 1.
• These variables are simply placeholders for numbers, allowing generalizations and simplifications.

Understanding these foundational concepts is vital when tackling more complex algebraic equations or expressions.