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Chapter 6: Problem 406

Divide the monomials. $$56 b^{8} \div 7 b^{2}$$

Short Answer

Expert verified
8 b^{6}

Step by step solution

01

Simplify the Coefficients

Divide the coefficients of the monomials. Divide 56 by 7 to get 8.
02

Subtract the Exponents

Subtract the exponents of the same base by applying the rule for dividing powers. For the base b: Subtract the exponent in the denominator (2) from the exponent in the numerator (8):
03

Combine the Results

Combine the simplified coefficient and the base with the new exponent. The result is: 8b^{6}

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Key Concepts

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

Simplifying Coefficients
When dividing monomials, the first step is to simplify the coefficients. Coefficients are the numbers in front of the variable(s). In the given example, we have 56 and 7 as coefficients.

To simplify them, divide the coefficient of the numerator (56) by the coefficient of the denominator (7):

\[ \frac{56}{7} = 8 \]

This step turns our monomial division into a simpler form, where we now only need to focus on the variable and its exponents. Always ensure you completely simplify the coefficients before moving on to the exponents.
Exponent Subtraction
Once the coefficients are simplified, we need to handle the exponents. When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Given: \( b^8 \bigg/ b^2 \), use the rule:

\[ a^m \bigg/ a^n = a^{m-n} \]

So, for our example:

\[ b^8 \bigg/ b^2 = b^{8-2}= b^6 \]

Always ensure you're subtracting the exponents correctly and are left with a positive exponent. This simplifies our term even further to \( b^6 \).
Algebraic Division
Combining all the parts is the final step. Now that we have simplified the coefficients and the variable, we combine them for the final result.

From the previous steps, we have:

\[ 8 \times b^6 \]

This is the simplified form of our original division problem:\(56 b^8 \bigg/ 7 b^2 \).

In algebraic division, breaking the problem into these simpler steps鈥攕implifying coefficients and subtracting exponents鈥攁llows us to effectively and accurately find the result.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\left(55 w^{2}-10 w\right) \div 5 w$$

Divide the monomials. $$\frac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$$

Simplify. (a) \(5^{-3}\) (b) \(10^{-5}\)

Divide each polynomial by the monomial. $$\frac{4 w^{2}+2 w-5}{2 w}$$

Divide each polynomial by the monomial. $$\left(8 x^{3}+6 x^{2}\right) \div 2 x$$
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