Problem 13 Divide. $$\frac{-36 b^{7}}{9 b^{... [FREE SOLUTION]

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Chapter 3: Problem 13

Divide. $$\frac{-36 b^{7}}{9 b^{5}}$$

Short Answer

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Question: Simplify the fraction $$\frac{-36 b^{7}}{9 b^{5}}.$$ Answer: $$-4b^{2}$$

Step by step solution

Divide the coefficients

To divide the coefficients, we simply divide -36 by 9: $$ \frac{-36}{9} = -4 $$

Divide the variables

Next, we will divide the variables with their corresponding exponents. Since both variables are the same (b), we will subtract their exponents: $$ \frac{b^{7}}{b^{5}} = b^{7-5} $$

Simplify the exponent

Now, we will simplify the exponent in the term we found in Step 2: $$ b^{7-5} = b^{2} $$

Combine the results

Finally, we will combine the results from Steps 1 and 3, which represent the coefficients and the variable: $$ -4b^{2} $$ So, the simplified fraction is: $$ \frac{-36 b^{7}}{9 b^{5}} = -4b^{2} $$

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Key Concepts

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

Exponent Rules

When dealing with exponents, one of the fundamental rules is how to divide them when they have the same base. The base is the number or variable that is being raised to a power. In our example, the base is the variable $b$.

To divide terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. In this exercise, you saw this as $b^7 \div b^5$, which becomes $b^{7-5}$ using the rule:

Quotient of Powers Rule: $ \frac{a^m}{a^n} = a^{m-n} $

The result is $b^2$ because you subtract 5 from 7.

This rule makes simplifying expressions with exponents much easier and is part of what makes polynomial division manageable.

Simplifying Fractions

Simplifying fractions is a core part of algebra that helps make expressions more manageable. It's all about reducing the fraction to its most basic form without changing its value.

In this exercise, you simplified the fraction by dividing both the coefficients and the variables separately:

First, you simplified $\frac{-36}{9}$ to $-4$. You did this by dividing the numerator and the denominator by their greatest common divisor, which is 9.
Second, you simplified the variables by applying the exponent rules, turning $\frac{b^7}{b^5}$ into $b^{2}$.

Combining these simplifications results in a simpler expression, $-4b^2$.

By breaking down the fraction like this, you can more easily understand and solve algebraic problems.

Polynomial Division

Polynomial division involves dividing two polynomial expressions, similar to how you would with numbers or simpler terms. The process involves both the coefficients and the variables.

For polynomial division, the concept is straightforward once you understand the rules for dividing coefficients and using exponent rules for like bases. In this example, you divided the polynomial terms $-36b^7$ by $9b^5$. This meant:

Dividing the coefficients $-36$ by $9$ to get $-4$.
Using the exponent rule $b^{7-5}$ to simplify the variable part.

After this division, you get the result $-4b^2$, which is the quotient of the original polynomial division.

These steps are critical when dealing with algebraic expressions and form the foundation of more complex polynomial division tasks you may encounter later.

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91影视

Divide. $$\frac{-36 b^{7}}{9 b^{5}}$$

Short Answer

Step by step solution

Divide the coefficients

Divide the variables

Simplify the exponent

Combine the results

Key Concepts

Exponent Rules

Simplifying Fractions

Polynomial Division

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