Math  /  Algebra

Questionstion Watch Video Show Examples ress the following fraction in simplest form using only positive exponents. ((4u2)412u5\frac{\left(\left(4 u^{2}\right)^{4}\right.}{12 u^{5}}

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with variables and exponents, making sure the final answer only has positive exponents. Watch out! Exponent rules can be tricky!
Don't mix up multiplying and adding exponents.
Also, remember to simplify both the numbers and the variables.

STEP 2

1. Expand the numerator
2. Simplify the numerator
3. Simplify the fraction

STEP 3

Alright, let's **expand** that numerator!
We have (4u2)4\left(4u^2\right)^4.
Remember, this means we're multiplying (4u2)\left(4u^2\right) by itself **four** times.
When we have an exponent outside parentheses, we **distribute** it to everything inside.
So, we get 44(u2)44^4 \cdot \left(u^2\right)^4.

STEP 4

444^4 means 44444 \cdot 4 \cdot 4 \cdot 4, which equals **256**.
Awesome!

STEP 5

Now, for the variable part: (u2)4\left(u^2\right)^4.
When we have a power raised to another power, we **multiply** the exponents.
So, (u2)4\left(u^2\right)^4 becomes u24=u8u^{2 \cdot 4} = u^8.

STEP 6

Putting it all together, our numerator becomes 256u8256u^8.

STEP 7

Our fraction now looks like this: 256u812u5\frac{256u^8}{12u^5}.
Much better than before, right?

STEP 8

We can **simplify** the numbers by dividing both 256 and 12 by their greatest common factor, which is 4.
So, 25612\frac{256}{12} becomes 256÷412÷4=643\frac{256 \div 4}{12 \div 4} = \frac{64}{3}.

STEP 9

Now, let's look at the variables: u8u5\frac{u^8}{u^5}.
When we divide terms with the same base, we **subtract** the exponents.
So, u8u5\frac{u^8}{u^5} becomes u85=u3u^{8-5} = u^3.

STEP 10

Finally, we combine our simplified number and variable to get our **final answer**: 643u3\frac{64}{3}u^3.

STEP 11

The simplified fraction is 643u3\frac{64}{3}u^3.

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