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Math  /  Algebra

QuestionPascal's Triangle to expand (3+2x2)4\left(3+2 x^{2}\right)^{4}. Express your answer in simplest form.

Studdy Solution

STEP 1

What is this asking? We need to expand the expression (3+2x2)4(3 + 2x^2)^4 using Pascal's Triangle, which gives us the coefficients for the expansion of a binomial raised to a power. Watch out! Don't forget to apply the powers to *both* the **coefficient** and the **variable** within each term of the binomial!
Also, remember to simplify the final expression by combining like terms.

STEP 2

1. Pascal's Triangle
2. Expand the Expression
3. Simplify the Terms
4. Combine Like Terms

STEP 3

Let's **build** Pascal's Triangle up to the 4th row (remember, the top row is row 0!). Each number is the sum of the two numbers directly above it. 111121133114641\begin{array}{ccccc} & & 1 & & \\ & 1 & & 1 & \\ 1 & & 2 & & 1 \\ 1 & 3 & & 3 & 1 \\ 1 & 4 & 6 & 4 & 1 \end{array}

STEP 4

The **4th row** (1, 4, 6, 4, 1) gives us the coefficients for our expansion!
How exciting is that?!

STEP 5

Now, we'll **expand** (3+2x2)4(3 + 2x^2)^4 using these **coefficients** and decreasing powers of 3 and increasing powers of 2x22x^2.
Fasten your seatbelts, here we go! 1⋅34(2x2)0+4⋅33(2x2)1+6⋅32(2x2)2+4⋅31(2x2)3+1⋅30(2x2)41\cdot 3^4(2x^2)^0 + 4\cdot 3^3(2x^2)^1 + 6\cdot 3^2(2x^2)^2 + 4\cdot 3^1(2x^2)^3 + 1\cdot 3^0(2x^2)^4

STEP 6

Let's **simplify** each term.
Remember to apply the powers to both the coefficient *and* the variable.
This is where we show off our exponent skills! 1⋅81⋅1+4⋅27⋅2x2+6⋅9⋅4x4+4⋅3⋅8x6+1⋅1⋅16x81\cdot 81\cdot 1 + 4\cdot 27\cdot 2x^2 + 6\cdot 9\cdot 4x^4 + 4\cdot 3\cdot 8x^6 + 1\cdot 1\cdot 16x^8

STEP 7

Further **simplification** gives us: 81+216x2+216x4+96x6+16x881 + 216x^2 + 216x^4 + 96x^6 + 16x^8

STEP 8

In this case, we don't have any like terms to combine.
That makes our job easier!
We already have our **simplified expanded expression**.

STEP 9

Our **final answer** is 81+216x2+216x4+96x6+16x881 + 216x^2 + 216x^4 + 96x^6 + 16x^8.
We expanded and simplified like true math superheroes!

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