Problem 60
Question
For exercises 39-82, simplify. $$ \frac{40 n^{5}}{21} \div 8 n $$
Step-by-Step Solution
Verified Answer
\[\frac{5 n^{4}}{21}\]
1Step 1: Write the Division as a Multiplication
Rewrite the division expression as a multiplication by the reciprocal. \[\frac{40 n^{5}}{21} \times \frac{1}{8 n}\]
2Step 2: Multiply the Numerators and Denominators
Multiply the numerators together and the denominators together. \[\frac{40 n^{5} \times 1}{21 \times 8 n}\]
3Step 3: Simplify the Fractions
Divide the numbers in the numerator and the denominator by their greatest common divisor. \[\frac{40 n^{5}}{168 n}\]
4Step 4: Reduce the Coefficients
Simplify the coefficients by dividing 40 and 168 by their greatest common divisor, which is 8. \[\frac{5 n^{5}}{21 n}\]
5Step 5: Simplify the Exponents
Reduce the exponents in the fraction by subtracting the exponent in the denominator from the exponent in the numerator. \[\frac{5 n^{4}}{21}\]
Key Concepts
Division of FractionsMultiplication of FractionsGreatest Common DivisorSimplifying Exponents
Division of Fractions
To divide one fraction by another, you need to multiply by the reciprocal of the fraction you're dividing by. The reciprocal simply means flipping the numerator and the denominator.
If you have \(\frac{a}{b} ÷ \frac{c}{d}\), you can rewrite this as \(\frac{a}{b} \times \frac{d}{c}\).
This transforms the division problem into a multiplication one, which is often easier to handle.
Remember to treat complex expressions carefully, ensuring each term is properly flipped when making the reciprocal.
If you have \(\frac{a}{b} ÷ \frac{c}{d}\), you can rewrite this as \(\frac{a}{b} \times \frac{d}{c}\).
This transforms the division problem into a multiplication one, which is often easier to handle.
Remember to treat complex expressions carefully, ensuring each term is properly flipped when making the reciprocal.
Multiplication of Fractions
Multiplying fractions involves a straightforward process. You multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
This rule also applies when fractions include variables and coefficients, such as in our given problem.
Combine all terms in the numerators and denominators before simplifying, ensuring all multiplications are complete before moving to the next step.
This rule also applies when fractions include variables and coefficients, such as in our given problem.
Combine all terms in the numerators and denominators before simplifying, ensuring all multiplications are complete before moving to the next step.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder.
For instance, the GCD of 40 and 168 is 8. To find the GCD, you can use several methods, including:
For instance, the GCD of 40 and 168 is 8. To find the GCD, you can use several methods, including:
- Prime factorization: Break down each number into prime factors and multiply the common factors.
- Euclidean algorithm: Repeatedly apply division and take remainders until reaching zero.
Simplifying Exponents
Simplifying exponents in algebraic expressions involves basic rules. When you divide terms with the same base, subtract the exponents.
The general rule is \(x^a / x^b = x^{a-b}\). Applying this to our expression, you reduce the exponent in the numerator by subtracting the exponent of the denominator.
In our example, \(n^{5}/ n\), we get \(n^{5-1}=n^{4}\). Always ensure the bases are the same before performing the subtraction.
The general rule is \(x^a / x^b = x^{a-b}\). Applying this to our expression, you reduce the exponent in the numerator by subtracting the exponent of the denominator.
In our example, \(n^{5}/ n\), we get \(n^{5-1}=n^{4}\). Always ensure the bases are the same before performing the subtraction.
Other exercises in this chapter
Problem 60
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{y}+\frac{1}{x}}{y+x} $$
View solution Problem 60
For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 12 a^{2} b ; 18 a b^{2} $$
View solution Problem 60
For exercises 1-66, simplify. $$ \frac{y^{3}-y^{2}-56 y}{y^{4}+5 y^{3}-14 y^{2}} $$
View solution Problem 61
For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake
View solution