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Chapter 1: Problem 30

Divide and simplify. $$ \frac{-20 x^{8} y^{5} z^{3}}{-4 x^{2} y^{2} z} $$

Short Answer

Expert verified
The simplified expression is \( 5x^{6}y^{3}z^{2} \).

Step by step solution

01

Write Down the Problem

The given problem is: \[ \frac{-20 x^{8} y^{5} z^{3}}{-4 x^{2} y^{2} z} \].
02

Simplify the Coefficients

Divide the coefficients: \( -20 \div -4 = 5 \). So, the expression simplifies to: \[ \frac{20 x^{8} y^{5} z^{3}}{4 x^{2} y^{2} z} = 5 x^{8} y^{5} z^{3} \].
03

Simplify the Variables with Exponents

Now the expression is simplified as: \[ 5 x^{6} y^{3} z^{2} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividing Polynomials
Dividing polynomials might seem tricky at first, but with a systematic approach, it becomes manageable. When you divide polynomials, you essentially break down a complex expression into simpler parts. Let's illustrate this with the example: \ \( \frac{-20 x^{8} y^{5} z^{3}}{-4 x^{2} y^{2} z} \). Here鈥檚 how you do it:
  • First, divide the coefficients (numerical parts) of the terms. In our example: \( -20 \div -4 = 5 \).
  • Next, divide the variables by subtracting the exponents of like bases. This rule comes from the property \( \frac{a^m}{a^n} = a^{m-n} \).
This will simplify our expression step by step until we reach the final answer. So, once you get the hang of it, dividing polynomials becomes straightforward!
Simplifying Exponents
Simplifying exponents is a crucial part of solving rational expressions. It involves using the laws of exponents effectively. Let's break this down:
  • When you have like bases, you can subtract the exponents for division: \( \frac{x^m}{x^n} = x^{m-n} \).
  • In our example: \( \frac{x^{8}}{x^{2}} = x^{8-2} = x^{6} \).
  • Apply this rule to all variables: \( \frac{y^{5}}{y^{2}} = y^{3} \) and \( \frac{z^{3}}{z} = z^{3-1} = z^{2} \).
By applying these rules consistently, you simplify your expression down to its simplest form. For our given problem: \( 5 x^{6} y^{3} z^{2} \). Remember, practice these steps, and soon enough they鈥檒l become second nature!
Rational Expressions
Rational expressions are fractions that involve polynomials both in the numerator and the denominator. Simplifying these expressions involves several key steps:
  • Factor both the numerator and the denominator, if possible. Since our initial expression \( \frac{-20 x^{8} y^{5} z^{3}}{-4 x^{2} y^{2} z} \) is already factored, we can move to the next steps.
  • Cancel out common factors in the numerator and the denominator.
  • Apply the division rules we discussed earlier to simplify exponents and coefficients.
Once we've implemented these steps, we're left with a simplified form. In our example, this results in \( 5 x^{6} y^{3} z^{2} \). Rational expressions might look complicated, but following these systematic steps can simplify even the complex ones!

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Most popular questions from this chapter

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ 4.3 \notin \mathbb{Z} $$

Divide. $$ \left(-\frac{2}{7}\right) \div(-1) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 32-(-5)^{2}+15 \div(-3) \cdot 2 $$

Calculate using the rules for order of operations. Are subtraction and division commutative? Why or why not?

Write an equivalent expression using an associative law. $$ -7(a b) $$
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