QuestionFactor completely: $(x+6)^{6}+(x+6)^{7}$
Answer Attempt 1 out of 3

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression with the greatest common factor pulled out. Watch out! Don't forget to factor completely!

STEP 2

1. Find the Greatest Common Factor (GCF).
2. Factor out the GCF.
3. Simplify the remaining expression.

STEP 3

Let's look at our expression: $(x+6)^{6}+(x+6)^{7}$ .
We've got two terms here, and they both have a common factor of $(x+6)$ raised to some power.

STEP 4

Remember, the greatest common factor is the largest thing that divides evenly into both terms.
In this case, it's $(x+6)$ raised to the smallest exponent, which is 6.
So, our GCF is $(x+6)^6$ .

STEP 5

Now, let's pull out that $(x+6)^6$ from both terms!
Remember, when we factor something out, we're essentially dividing each term by the GCF.

STEP 6

So, we have $(x+6)^6 \cdot \frac{(x+6)^6}{(x+6)^6} + (x+6)^6 \cdot \frac{(x+6)^7}{(x+6)^6}$ .

STEP 7

Let's simplify those fractions!
Remember, anything divided by itself is 1, so $\frac{(x+6)^6}{(x+6)^6} = 1$ .

STEP 8

For the second fraction, $\frac{(x+6)^7}{(x+6)^6}$ , we can use our exponent rules!
When we divide terms with the same base, we subtract the exponents.
So, $7-6=1$ , and we're left with $(x+6)^1$ , which is just $(x+6)$ .

STEP 9

Now, putting it all together, we have $(x+6)^6(1 + (x+6))$ .

STEP 10

We can simplify the expression inside the parentheses: $1 + (x+6) = x + 7$ .

STEP 11

So, our completely factored expression is $(x+6)^6(x+7)$ .
Awesome!

STEP 12

The completely factored form of $(x+6)^{6}+(x+6)^{7}$ is $(x+6)^6(x+7)$ .

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