Problem 283 In the following exercises, divi... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 283

In the following exercises, divide the monomials. $$\frac{-35 x^{7}}{-42 x^{13}}$$

Short Answer

Expert verified

$\frac{5}{6 x^{6}}$

Step by step solution

Identify like terms in the monomials

To simplify $\frac{-35 x^{7}}{-42 x^{13}}$, start by identifying like terms in the numerator and the denominator. In this case, the coefficients are $-35$ and $-42$, and the variables are $x^7$ and $x^{13}$.

Simplify the coefficients

Divide the coefficients $-35$ by $-42$. The negatives cancel out, leaving $\frac{35}{42}$. Simplify by finding the greatest common divisor of 35 and 42, which is 7: $\frac{35 \div 7}{42 \div 7} = \frac{5}{6}$.

Simplify the variable terms

For the variables, divide the exponents by subtracting the exponent in the denominator from the exponent in the numerator: $x^{7-13} = x^{-6}$.

Combine simplified coefficients and variable terms

Combine the simplified coefficient $\frac{5}{6}$ with $x^{-6}$ to get $\frac{5}{6} x^{-6}$.

Convert negative exponent

Rewrite the expression with a positive exponent: $\frac{5}{6} x^{-6} = \frac{5}{6 x^{6}}$.

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Key Concepts

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

Simplifying Coefficients

Before diving into the main details of dividing monomials, let's start with simplifying the coefficients. In the problem provided, we need to simplify $-35$ and $-42$.

When we divide $-35$ by $-42$, the two negative signs cancel each other out, turning the division into a simpler one: \[\frac{35}{42}\].
The next step is to simplify \[\frac{35}{42}\].
Here, we use the greatest common divisor (GCD) of 35 and 42, which is 7. After dividing both numbers by 7, we get:

\[\frac{35 \div 7}{42 \div 7} \] simplifies to \[ \frac{5}{6} \]

This results in a simplified coefficient of \[\frac{5}{6}\].

Subtracting Exponents

Now that we have our simplified coefficient, let's move on to the exponents. Exponent rules are crucial when dividing monomials. Here, we're working with the expression $\frac{x^7}{x^{13}}$.
To simplify this, we use the rule of exponents that states:

When dividing like bases, we subtract the exponent in the denominator from the exponent in the numerator.

Applying this rule:
\[ x^{7-13} = x^{-6} \] The result is \ x^{-6} \ which we now have as part of our simplified expression.

Negative Exponents

Finally, let's discuss negative exponents. Having a negative exponent in our final expression means that we'll need to convert it to a positive one.
Our simplified monomial from before was: \[ \frac{5}{6} x^{-6} \]
According to the rules of exponents, \ a^{-n} = \frac{1}{a^n} \. Here's how we convert \ x^{-6} \ to a positive exponent:

\ x^{-6} \ becomes \ \frac{1}{x^6} \

Hence, combining our coefficient with the changed exponent, we transform \[ \frac{5}{6} x^{-6} \] to: \[ \frac{5}{6} \frac{1}{x^6} \]

Which finally simplifies to \[ \frac{5}{6 x^6} \]
And there we have it!
This is your simplified expression.

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91影视

In the following exercises, divide the monomials. $$\frac{-35 x^{7}}{-42 x^{13}}$$

Short Answer

Step by step solution

Identify like terms in the monomials

Simplify the coefficients

Simplify the variable terms

Combine simplified coefficients and variable terms

Convert negative exponent

Key Concepts

Simplifying Coefficients

Subtracting Exponents

Negative Exponents

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