Problem 32 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{12 x^{-3} y^{-2} z^{-8}}{30 x^{-6} y^{-4} z^{-1}} $$

Step-by-Step Solution

Answer

\( \frac{2x^3 y^2}{5z^7} \).

Step 1: Simplify the Coefficients

First, simplify the numeric coefficients by dividing 12 by 30. Simplifying gives us \( \frac{12}{30} = \frac{2}{5} \).

Step 2: Simplify the Exponents for Each Variable

Apply the rule of exponents \( \frac{a^m}{a^n} = a^{m-n} \): - For \( x \): Subtract the exponents \(-3 - (-6) = 3\). - For \( y \): Subtract the exponents \(-2 - (-4) = 2\). - For \( z \): Subtract the exponents \(-8 - (-1) = -7\).

Step 3: Write the Simplified Expression

Combine the simplified coefficients and exponents: \[ \frac{2}{5}x^3 y^2 z^{-7} \] Since \( z^{-7} \) can be written as \( \frac{1}{z^7} \), the expression becomes: \[ \frac{2x^3 y^2}{5z^7} \].

Key Concepts

Rules of Exponents

Exponents can seem tricky at first, but understanding their rules makes simplifying expressions a lot easier! One of the most important rules is how to handle division. When dividing like bases, you simply subtract the exponents. This rule is written as \( \frac{a^m}{a^n} = a^{m-n} \). For example, if you have \( x^{-3} \) over \( x^{-6} \), you subtract \(-3\) from \(-6\), resulting in \( 3 \) because \(-3 - (-6) = 3\).
Another important rule is when dealing with zero or negative exponents. The rule \( a^0 = 1 \) tells us any number with an exponent of zero equals 1, while a negative exponent, like \( a^{-1} \), means \( \frac{1}{a} \). Remember, negative exponents never actually make numbers negative; instead, they help to express reciprocals!

Simplifying Coefficients

Simplifying coefficients helps to make complicated expressions look cleaner and easier to solve. Coefficients are the numeric part of any term. In our exercise, we have the coefficient 12 in the numerator and 30 in the denominator. Simplifying these coefficients is similar to simplifying fractions. You divide both numbers by their greatest common factor (GCF).
For this particular exercise, the GCF of 12 and 30 is 6. So, dividing both by 6 gives you \( \frac{12}{30} = \frac{2}{5} \).

• Think of simplifying as finding an equivalent fraction with the smallest numbers possible.
• After simplifying, you can focus more on the variables and their exponents.

This is just like reducing fractions in arithmetic but with the added fun of algebra!

Negative Exponents

Negative exponents might look intimidating, but they hold their own charm in algebraic simplification. They essentially tell us to "flip" the base to the other side of the fraction. For example, \( x^{-3} \) is equal to \( \frac{1}{x^3} \).
Applying this to the exercise, when we simplified the expression, we ended up with \( z^{-7} \), which is equivalent to \( \frac{1}{z^7} \). This transformation helps move terms from the numerator to the denominator or vice versa, simplifying the expression in a rational way.

• Remember: negative exponents switch positions between the numerator and the denominator.
• They make expressions more manageable by converting terms to positive exponents.

Understanding negative exponents is key to mastering algebraic expressions and realizing their potential to simplify complex equations.