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.

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions

Struggling with Intermediate Algebra?

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Multiple Choice

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

$x^{4}$

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

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

$4x^6$

Verified step by step guidance

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.

3:41m

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Struggling with Intermediate Algebra?

Divide each expression and write the quotient in its simplest form.8x3(2x)5÷4x416x2\(\frac{8x^3}{(2x)^5}\[\div\]\frac{4x^4}{16x^2}\)

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