Problem 302 $$\left(8 x^{5}\right)(9 x) \div... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 302

$$\left(8 x^{5}\right)(9 x) \div 6 x^{3}$$

Short Answer

Expert verified

12x^3

Step by step solution

Simplify the Multiplication

First, simplify the multiplication part $8x^5 \times 9x$. Multiply the coefficients and add the exponents of the same base (x): $8 \times 9 = 72$ and $x^5 \times x^1 = x^{5+1} = x^6$. So, $8x^5 \times 9x = 72x^6$.

Simplify the Division

Next, divide the result by $6x^3$. This involves dividing the coefficients and subtracting the exponents of the same base (x): \[\frac{72}{6} = 12\] and \[\frac{x^6}{x^3} = x^{6-3} = x^3\]. Thus, \[\frac{72x^6}{6x^3} = 12x^3\].

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Key Concepts

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

Multiplication of Polynomials

When you multiply polynomials, you combine the like terms by multiplying their coefficients and adding their exponents.
This process may seem a bit complex at first, but once you understand the basic rules, it becomes much easier.

For example, in the given exercise, we need to multiply $8x^5$ and $9x$.
Here鈥檚 how we do it:

Multiply the coefficients: $8 \times 9 = 72$
Add the exponents of the same base (x): $x^5 \times x^1 = x^{5+1} = x^6$

As a result, $8x^5 \times 9x = 72x^6$.
This combination of multiplication of coefficients and addition of exponents is a crucial part of polynomial multiplication.

Division of Polynomials

Dividing polynomials involves separating the coefficients and subtracting the exponents of the same base.
Similar to multiplication, this process follows specific rules to simplify the given expression.

Continuing from our example, we have $72x^6$, and we need to divide it by $6x^3$:

Divide the coefficients: $72 \div 6 = 12$
Subtract the exponents: $x^6 \div x^3 = x^{6-3} = x^3$

This gives us the simplified result: $ \frac{72x^6}{6x^3} = 12x^3$.
Understanding this allows you to handle polynomial division easily, as long as you keep track of how to split the coefficients and handle the exponents.

Laws of Exponents

The laws of exponents are fundamental rules that simplify the manipulation of expressions involving powers.
These laws include multiplication, division, and power of exponents.

Here are some essential rules:

Multiplication: $x^a \times x^b = x^{a+b}$
Division: $ \frac{x^a}{x^b} = x^{a-b} $
Power of a power: $ (x^a)^b = x^{a \times b} $

In the given exercise, we applied these laws:

During multiplication: $ x^5 \times x^1 = x^{5+1} = x^6 $

During division: $ x^6 \div x^3 = x^{6-3} = x^3 $

Applying these laws correctly simplifies complex expressions and is a critical skill in algebra.

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91影视

$$\left(8 x^{5}\right)(9 x) \div 6 x^{3}$$

Short Answer

Step by step solution

Simplify the Multiplication

Simplify the Division

Key Concepts

Multiplication of Polynomials

Division of Polynomials

Laws of Exponents

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