Divide each expression and write the quotient in its simplest form. 8 x 3 ( 2 x ) 5 ÷ 4 x 4 16 x 2 \frac{8x^3}{(2x)^5}\div\frac{4x^4}{16x^2}

So here we have a problem where we're told to divide each expression and write the quotient in its simplest form. We have eight x cubed over two x to the fifth power divided by four x to the fourth power over 16 x squared. Now before I do anything, I actually want to make this a bit more simple by completely writing out the expanded version of two x to the fifth power, since all of that is in parentheses. So we have eight x to the third on top. Now what I'm going to do is raise two and x to the fifth power. Two raised to the fifth power is 32, and then we will just have x to the fifth power. So that's what this looks like when we apply that exponent to the two and the x. Now this is going to be divided by everything that we have in this next fraction. But remember the rule when it comes to dividing fractions or dividing rational expressions. We can use keep, change, flip. So we keep the first fraction, we change the division to multiplication, and then we flip the second piece, which is gonna be 16 x squared over four x to the fourth power. So now we've gone ahead and we've found the reciprocal, in which case we can just multiply across. So what I'm going to end up with is eight times 16 is gonna be times x cubed times x squared. And I'm intentionally leaving this all in its expanded form, so that we can find common factors in the numerator and denominator. Now we have 32, it's going to be x to the fifth. And actually, I'm gonna write it so all of these numbers are our friends. We have 32 times four, it's gonna be times x to the fifth, and then times x to the fourth. Now what I need to do is figure out how to go ahead and simplify this. Now something that I know is that on top we'll have this eight, and then 16 is the same thing as four times four. And then in the denominator, I'm going to multiply out these constants or expand them. So 32 is the same thing as eight times four. Right? Eight times four is 32, and then I'm gonna go ahead and keep that four there as well. And then I'm going to have x cubed on top, and we're gonna have times x squared, and then we have x to the fifth, and times x to the fourth. Now what do I do from here? Well, what I notice is something really convenient happens, all of these constants just cancel, the eight times four times four completely cancel. So now I'm just left with the x's. So I'm left with x to the third times x squared, which I'm which I can also write as x times x times x times x times x. Right? Because we're going to have one, two, three, four, five x's because three and then two x's there, that gives us five x's being multiplied. And then in the denominator, we'll have x to the fifth times x to the fourth, so it's going to be nine x's total. Five plus four is nine. So we're gonna have one, two, three, four, five, six, seven, eight, and then we have nine x's right there. And what I can do is go ahead and cancel five of the x's in the numerator with five of the x's in the denominator, leaving us with just four x's in the denominator left over. So when everything is said and done and everything is simplified, we're just going to have a one on top. There's an implicit one being multiplied there, and we're gonna have four x's being multiplied, which is x to the fourth power. So the most simplified final answer to this problem is one over x to the fourth. As you can see, it can oftentimes take a lot of steps to go ahead and simplify things once you've gone ahead and multiply these across. But the main thing you need to remember, the main rule here is to keep the change and the flip. And this is how you can go about dividing any rational expressions that you see. Hope you found this helpful.

Indice

Skip topic navigation

1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
38m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

12. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

13. Sequences, Series, and the Binomial Theorem1h 50m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions

Problema precedenteProssimo problema

Difficoltà con Intermediate Algebra?

Unisciti a migliaia di studenti che si affidano a noi per superare al meglio i loro esami!

Scelta multipla

Divide each expression and write the quotient in its simplest form.
$\frac{8 x^{3}}{(2 x)^{5}} \div \frac{4 x^{4}}{16 x^{2}}$

$x^{4}$

$\frac{1}{x^{4}}$

$\frac{1}{4x^6}$

$4x^6$

0 Commenti

Problema precedenteProssimo problema

Guida verificata passo dopo passo

Rewrite the division of fractions as multiplication by the reciprocal. The original expression is: $\frac{8x^3}{(2x)^5} \div \frac{4x^4}{16x^2}$. This becomes: $\frac{8x^3}{(2x)^5} \times \frac{16x^2}{4x^4}$.

Simplify the powers and constants inside the fractions. First, expand the denominator $(2x)^5$ as $2^5 \cdot x^5 = 32x^5$. So the first fraction becomes $\frac{8x^3}{32x^5}$.

Simplify each fraction by dividing the coefficients and subtracting exponents of like bases. For $\frac{8x^3}{32x^5}$, divide 8 by 32 and subtract exponents of $x$: $x^{3-5} = x^{-2}$. For the second fraction $\frac{16x^2}{4x^4}$, divide 16 by 4 and subtract exponents of $x$: $x^{2-4} = x^{-2}$.

Multiply the simplified fractions: multiply the coefficients and multiply the powers of $x$ by adding exponents. So multiply the coefficients from step 3 and add the exponents of $x$ from both fractions.

Express the final answer with positive exponents by moving terms with negative exponents to the denominator. This will give the quotient in its simplest form.