Problem 6 Multiply. $$\left(8 p^{6}\right... [FREE SOLUTION]

Chapter 6: Problem 6

Multiply. $$\left(8 p^{6}\right)\left(-p^{4}\right)$$

Short Answer

Expert verified

The short version of the answer is: $-8p^{10}$.

Step by step solution

Multiply the coefficients

Multiply the coefficients of the two monomials: $8$ and $-1$ (considering the '-' sign as a coefficient) $8 \times -1 = -8$

Add the exponents of the variables

Add the exponents of the variables with the same base: $p^{6} \times p^{4} = p^{(6 + 4)} = p^{10}$

Combine the products

Combine the products of coefficients and variables in a single expression: $-8p^{10}$ So, the result of multiplying $(8p^6) \times (-p^4)$ is: $-8p^{10}$

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Key Concepts

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

Monomials

A monomial is the simplest type of algebraic expression, containing only one term. This term is formed by multiplying a constant with one or more variables. Each variable in a monomial has a non-negative integer exponent.
For instance, in the exercise, we have two monomials: $8p^6$ and $-p^4$. Here, 8 is the coefficient, and $p^6$ and $p^4$ are the variable parts.

The number 8 and -1 (considering the minus sign as part of the number) are the coefficients of the monomials.
The powers $p^6$ and $p^4$ indicate the variable 'p' raised to an exponent.

Remember, a monomial does not contain addition or subtraction within the term, making it a straightforward expression. Understanding monomials helps in performing operations like multiplication with ease.

Exponents Addition

In arithmetic involving exponents, specifically in the multiplication of similar bases, we add the exponents together. This is a shorthand since multiplying bases of the same variable involves combining their powers.
Let's take the exercise example: $p^6$ and $p^4$. When these are multiplied

We add the exponents: $6 + 4$.
The result is $p^{10}$.

This step in dealing with exponents is crucial as it simplifies expressions swiftly. Knowing this rule helps prevent mistakes and speeds up calculations when dealing with larger algebraic expressions.

Coefficients Multiplication

When multiplying monomials, coefficients (the numerical parts) are handled just like regular numbers. You multiply them separately from the variables. This clear separation ensures accuracy.

For example, in our exercise, we have:

The coefficients 8 (from $8p^6$) and -1 (from $-p^4$).
We perform the operation: $8 \times -1 = -8$.

The result, -8, becomes the new coefficient in our expression. After combining this with the variables, we get the complete expression $-8p^{10}$. Keeping operations separate for coefficients and variables simplifies your work and reduces errors in polynomial multiplication.

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91影视

Multiply. $$\left(8 p^{6}\right)\left(-p^{4}\right)$$

Short Answer

Step by step solution

Multiply the coefficients

Add the exponents of the variables

Combine the products

Key Concepts

Monomials

Exponents Addition

Coefficients Multiplication

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