Math  /  Algebra

Questionc) (2x2)3(2x)2\frac{\left(2 x^{2}\right)^{3}}{(2 x)^{2}}

Studdy Solution

STEP 1

What is this asking? Simplify this big fraction of xx terms by applying exponent rules. Watch out! Don't mix up the exponents inside and outside the parentheses!
And remember, everything inside the parentheses gets the exponent outside.

STEP 2

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

STEP 3

Alright, let's **expand** that numerator (2x2)3(2x^2)^3!
Remember, *everything* inside those parentheses gets the exponent outside.
So, we have (2x2)3=23(x2)3(2x^2)^3 = 2^3 \cdot (x^2)^3.
We're doing this because each term inside the parenthesis is multiplied by itself three times!

STEP 4

Now, 232^3 is just 222=82 \cdot 2 \cdot 2 = \textbf{8}.
And (x2)3(x^2)^3 is x^{2 \cdot 3} = \textbf{x^6}.
We multiply the exponents together because it's like we have three sets of two xx’s multiplied together.
So, our numerator becomes \textbf{8x^6}.

STEP 5

Now for the denominator, (2x)2(2x)^2.
Same rule applies: *everything* inside gets the exponent.
So, (2x)2=22x2(2x)^2 = 2^2 \cdot x^2.

STEP 6

222^2 is 22=42 \cdot 2 = \textbf{4}, and x2x^2 is just, well, \textbf{x^2}.
So, our denominator becomes \textbf{4x^2}.

STEP 7

Now, let's put it all together!
We have 8x64x2\frac{8x^6}{4x^2}.

STEP 8

We can simplify this by dividing the numbers and then the xx terms. 88 divided by 44 is 2\textbf{2}.
So, we have 2x6x22 \cdot \frac{x^6}{x^2}.

STEP 9

Now, \frac{x^6}{x^2} = x^{6-2} = \textbf{x^4}.
We subtract the exponents because we're essentially dividing out two xx’s from the top and bottom, leaving four xx’s on top.

STEP 10

Putting it all together, we get \textbf{2x^4}!

STEP 11

The simplified expression is 2x42x^4.

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