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What value of [tex]n[/tex] makes the statement true?

[tex]6x^n \cdot 4x^2 = 24x^6[/tex]

What value of [tex]c[/tex] makes the statement true?

[tex]-2x^3(cx^3 + x^2) = -10x^6 - 2x^5[/tex]

Answer :

Let's solve for the value of [tex]\( c \)[/tex] that makes the given statement true.

The equation we have is:

[tex]\[
-2x^3\left(c x^3 + x^2\right) = -10x^6 - 2x^5
\][/tex]

### Step 1: Distribute the [tex]\(-2x^3\)[/tex]

First, distribute [tex]\(-2x^3\)[/tex] across the terms inside the parentheses on the left side of the equation:

[tex]\[
-2x^3 \cdot (c x^3) + (-2x^3 \cdot x^2)
\][/tex]

This gives:

[tex]\[
-2c x^6 - 2x^5
\][/tex]

### Step 2: Set the expressions equal

Now, compare the left-hand side [tex]\(-2c x^6 - 2x^5\)[/tex] with the right-hand side [tex]\(-10x^6 - 2x^5\)[/tex]:

[tex]\[
-2c x^6 - 2x^5 = -10x^6 - 2x^5
\][/tex]

### Step 3: Compare coefficients

We see that for the terms involving [tex]\( x^6 \)[/tex], the coefficients must be equal. So, set:

[tex]\[
-2c = -10
\][/tex]

### Step 4: Solve for [tex]\( c \)[/tex]

Divide both sides of the equation by [tex]\(-2\)[/tex] to solve for [tex]\( c \)[/tex]:

[tex]\[
c = \frac{-10}{-2}
\][/tex]

### Conclusion

Thus, the value of [tex]\( c \)[/tex] that makes the statement true is [tex]\( c = 5 \)[/tex].

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