Problem 56
Question
For exercises 39-82, simplify. $$ \frac{5 b^{2}+10 b}{8} \div \frac{b+2}{2 b^{3}} $$
Step-by-Step Solution
Verified Answer
\[ \frac{5b^4}{4} \]
1Step 1: Rewrite Division as Multiplication
Rewrite the division of fractions as multiplication by the reciprocal. The given expression is: \[ \frac{5b^2 + 10b}{8} \times \frac{2b^3}{b + 2} \]
2Step 2: Factor the Numerator
Factor the numerator of the first fraction, \( 5b^2 + 10b \), as follows: \[ 5b^2 + 10b = 5b(b + 2) \]
3Step 3: Simplify the Multiplication
Substitute the factored form into the multiplication expression: \[ \frac{5b(b + 2)}{8} \times \frac{2b^3}{b + 2} \]
4Step 4: Cancel Common Factors
Cancel the \( b + 2 \) terms and simplify the coefficients: \[ \frac{5b}{8} \times 2b^3 \]
5Step 5: Multiply the Remaining Fractions
Multiply the remaining fractions together: \[ \frac{5b \times 2b^3}{8} = \frac{10b^4}{8} \]
6Step 6: Simplify the Final Expression
Simplify the fraction \( \frac{10b^4}{8} \) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \[ \frac{10b^4}{8} = \frac{5b^4}{4} \]
Key Concepts
factoring polynomialsmultiplying fractionscanceling common factorssimplifying fractions
factoring polynomials
Factoring polynomials is a crucial step in simplifying algebraic expressions. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial. In this exercise, we need to factor the numerator of the first fraction, which is given as \(5b^2 + 10b\).
Recognize common factors: Look for common terms in each term of the polynomial. Here, both terms of \(5b^2 + 10b\) have a common factor of \(5b\).
Factor out the common term: When we factor out \(5b\) from \(5b^2 + 10b\), we get: \[ 5b^2 + 10b = 5b(b + 2) \]
This simpler form will be helpful when simplifying the entire fraction.
Recognize common factors: Look for common terms in each term of the polynomial. Here, both terms of \(5b^2 + 10b\) have a common factor of \(5b\).
Factor out the common term: When we factor out \(5b\) from \(5b^2 + 10b\), we get: \[ 5b^2 + 10b = 5b(b + 2) \]
This simpler form will be helpful when simplifying the entire fraction.
multiplying fractions
Multiplying fractions involves multiplying the numerators together and the denominators together.
Consider two fractions: \(\frac{a}{b}\) and \(\frac{c}{d}\). When we multiply them, the result is: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
In our exercise, we first convert the division problem into a multiplication problem by taking the reciprocal of the second fraction: \( \frac{5b^2 + 10b}{8} \div \frac{b + 2}{2b^3} \) becomes \( \frac{5b^2 + 10b}{8} \times \frac{2b^3}{b + 2} \).
After factoring, we have \( \frac{5b(b+2)}{8} \times \frac{2b^3}{b+2} \).
Now, we can multiply the fractions directly and simplify.
Consider two fractions: \(\frac{a}{b}\) and \(\frac{c}{d}\). When we multiply them, the result is: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
In our exercise, we first convert the division problem into a multiplication problem by taking the reciprocal of the second fraction: \( \frac{5b^2 + 10b}{8} \div \frac{b + 2}{2b^3} \) becomes \( \frac{5b^2 + 10b}{8} \times \frac{2b^3}{b + 2} \).
After factoring, we have \( \frac{5b(b+2)}{8} \times \frac{2b^3}{b+2} \).
Now, we can multiply the fractions directly and simplify.
canceling common factors
Canceling common factors is a method to simplify an expression by removing identical terms from the numerator and denominator.
Look for common terms: Identify terms in one fraction's numerator that also appear in the other fraction's denominator.
In the expression \( \frac{5b(b+2)}{8} \times \frac{2b^3}{b+2} \), we see \( b+2 \) appears in both.
Cancel these common factors, resulting in: \( \frac{5b}{8} \times 2b^3 \)
Look for common terms: Identify terms in one fraction's numerator that also appear in the other fraction's denominator.
In the expression \( \frac{5b(b+2)}{8} \times \frac{2b^3}{b+2} \), we see \( b+2 \) appears in both.
Cancel these common factors, resulting in: \( \frac{5b}{8} \times 2b^3 \)
simplifying fractions
Simplifying fractions involves reducing them to their simplest form.
Multiply remaining terms: After canceling common factors, we have: \( \frac{5b}{8} \times 2b^3 \). Multiply the numerators and denominators: \( \frac{5b \times 2b^3}{8} \), resulting in \( \frac{10b^4}{8} \).
Finally, reduce the fraction: Divide both numerator and denominator by their greatest common divisor (GCD).
Here, the GCD is 2.
Thus, the simplified form is: \( \frac{10b^4}{8} = \frac{5b^4}{4} \).
Now, the algebraic expression is in its simplest form.
Multiply remaining terms: After canceling common factors, we have: \( \frac{5b}{8} \times 2b^3 \). Multiply the numerators and denominators: \( \frac{5b \times 2b^3}{8} \), resulting in \( \frac{10b^4}{8} \).
Finally, reduce the fraction: Divide both numerator and denominator by their greatest common divisor (GCD).
Here, the GCD is 2.
Thus, the simplified form is: \( \frac{10b^4}{8} = \frac{5b^4}{4} \).
Now, the algebraic expression is in its simplest form.
Other exercises in this chapter
Problem 55
For exercises 39-82, simplify. $$ \frac{2 a^{2}+2 a}{9} \div \frac{a+1}{3 a^{2}} $$
View solution Problem 56
For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 15 ; 20 $$
View solution Problem 56
For exercises 1-66, simplify. $$ \frac{10 a^{2}+20 a-30}{5 a^{2}+20 a+15} $$
View solution Problem 57
When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where
View solution