Question

40 points and brainliest to the best correct answer, questions in photo.

Answer 1

Answer:

2. w = -3

4. c = -2

6. r = 7

8. n = 4

Step-by-step explanation:

It's hard to read the photo, but I'll try.

2.

$2^{w + 4} \times 2^{4w + 6} = 2^{2w + 1}$

On the left side, you have a product of 2 powers with the same base.

Give the same base and add the exponents, as in the rule

$a^m \times a^n = a^{m + n}$

You get on the left side 2 to the sum of the exponents.

$2^{w + 4 + 4w + 6} = 2^{2w + 1}$

Simplify the long exponent on the left side.

$2^{5w + 10} = 2^{2w + 1}$

Now we have another rule we can use. If two powers are equal, and their bases are equal, then the exponents must be equal.

Here, 2 to the power 5w + 10 equals 2 to the power 2w + 1. Since both bases are 2 and are equal, then the exponents must be equal.

5w + 10 = 2w + 1

Subtract 2w from both sides. Subtract 10 from both sides.

3w = -9

Divide both sides by 3.

w = -3

4.

$\dfrac{1}{5} = 5^{2c + 3}$

We need to write the left side as a power of 5. Then we equate the exponents like we did in problem 2.

Recall the rule:

$a^{-n} = \dfrac{1}{n}$

Apply this rule to the left side in reverse.

$5^{-1} = 5^{2c + 3}$

Now we have two powers that are equal, both having the same base, 5, so the exponents must be equal.

2c + 3 = -1

2c = -4

c = -2

6.

$216 = 6^{2r - 11}$

This is the same idea as problem 4. Write the left side as a power of 6.

$6^3 = 6^{2r - 11}$

$2r - 11 = 3$

$2r = 14$

$r = 7$

r = 7

8.

$4^{n} \times 4^{2n - 9} = 64$

This problem is similar to problem 2.

$4^{n + 2n - 9} = 4^3$

$4^{3n - 9} = 4^3$

3n - 9 = 3

3n = 12

n = 4