Math  /  Algebra

Question10. Simplify: (2x4y2x)3\left(\frac{2 x^{-4}}{y^{-2} x}\right)^{-3} (a) 6x15y6-\frac{6 x^{15}}{y^{6}} (b) 8x15y6-\frac{8 x^{15}}{y^{6}} (c) 6y5x4-\frac{6 y^{5}}{x^{4}} (d) y58x4\frac{y^{5}}{8 x^{4}} (e) x158y6\frac{x^{15}}{8 y^{6}}

Studdy Solution

STEP 1

What is this asking? We need to simplify a funky fraction with negative exponents, and then pick the right simplified version from the choices! Watch out! Negative exponents can be tricky, so let's be super careful with those and not mix up the rules!

STEP 2

1. Simplify inside the parentheses.
2. Apply the outer exponent.
3. Tidy up and finalize.

STEP 3

Alright, let's look inside those parentheses first.
We've got x4x^{-4} and xx hanging out together.
Remember, xx is the same as x1x^1.
When we divide terms with the same base, we subtract the exponents.
So, x4/x1=x41=x5x^{-4} / x^1 = x^{-4-1} = x^{-5}.
Inside the parentheses, we now have 2x5y2\frac{2x^{-5}}{y^{-2}}.

STEP 4

Let's get rid of those negative exponents inside the parentheses.
Remember, xn=1xnx^{-n} = \frac{1}{x^n} and 1yn=yn\frac{1}{y^{-n}} = y^n.
So, x5x^{-5} moves to the bottom as x5x^5, and y2y^{-2} moves to the top as y2y^2.
Now, we have 2y2x5\frac{2y^2}{x^5} inside the parentheses.

STEP 5

Now, we have (2y2x5)3\left(\frac{2y^2}{x^5}\right)^{-3}.
Let's distribute that 3-3 exponent to every term inside: 232^{-3}, (y2)3(y^2)^{-3}, and (x5)3(x^5)^{-3}.

STEP 6

Remember, (am)n=amn(a^m)^n = a^{m \cdot n}.
So, (y2)3=y2(3)=y6(y^2)^{-3} = y^{2 \cdot (-3)} = y^{-6}, and (x5)3=x5(3)=x15(x^5)^{-3} = x^{5 \cdot (-3)} = x^{-15}.
Also, 23=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}.
Our expression now looks like 18y6x15\frac{1}{8} \cdot y^{-6} \cdot x^{-15}.

STEP 7

Let's move those negative exponents again! y6y^{-6} goes to the bottom as y6y^6, and x15x^{-15} goes to the top as x15x^{15}.
We now have x158y6\frac{x^{15}}{8y^6}.

STEP 8

Look at us go!
Our simplified expression is x158y6\frac{x^{15}}{8y^6}.

STEP 9

Our final answer is x158y6\frac{x^{15}}{8y^6}, which matches option (e)!

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord