Answer :
To solve the equation [tex]\(8^y = 16^{y+2}\)[/tex], let's follow these steps:
1. Express both bases as powers of 2:
- 8 can be written as [tex]\(2^3\)[/tex], so [tex]\(8^y\)[/tex] can be written as [tex]\((2^3)^y\)[/tex] which simplifies to [tex]\(2^{3y}\)[/tex].
- 16 can be written as [tex]\(2^4\)[/tex], so [tex]\(16^{y+2}\)[/tex] can be written as [tex]\((2^4)^{y+2}\)[/tex] which simplifies to [tex]\(2^{4(y+2)}\)[/tex].
2. Rewrite the equation with these expressions:
[tex]\[ 8^y = 16^{y+2} \implies 2^{3y} = 2^{4(y+2)} \][/tex]
3. Set the exponents equal to each other since the bases are the same:
[tex]\[ 3y = 4(y + 2) \][/tex]
4. Solve for [tex]\( y \)[/tex]:
- Distribute the 4 on the right-hand side:
[tex]\[ 3y = 4y + 8 \][/tex]
- To isolate [tex]\( y \)[/tex], subtract [tex]\( 4y \)[/tex] from both sides:
[tex]\[ 3y - 4y = 8 \implies -y = 8 \][/tex]
- Divide both sides by -1:
[tex]\[ y = -8 \][/tex]
So, the value of [tex]\( y \)[/tex] is [tex]\(-8\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{-8} \][/tex]
1. Express both bases as powers of 2:
- 8 can be written as [tex]\(2^3\)[/tex], so [tex]\(8^y\)[/tex] can be written as [tex]\((2^3)^y\)[/tex] which simplifies to [tex]\(2^{3y}\)[/tex].
- 16 can be written as [tex]\(2^4\)[/tex], so [tex]\(16^{y+2}\)[/tex] can be written as [tex]\((2^4)^{y+2}\)[/tex] which simplifies to [tex]\(2^{4(y+2)}\)[/tex].
2. Rewrite the equation with these expressions:
[tex]\[ 8^y = 16^{y+2} \implies 2^{3y} = 2^{4(y+2)} \][/tex]
3. Set the exponents equal to each other since the bases are the same:
[tex]\[ 3y = 4(y + 2) \][/tex]
4. Solve for [tex]\( y \)[/tex]:
- Distribute the 4 on the right-hand side:
[tex]\[ 3y = 4y + 8 \][/tex]
- To isolate [tex]\( y \)[/tex], subtract [tex]\( 4y \)[/tex] from both sides:
[tex]\[ 3y - 4y = 8 \implies -y = 8 \][/tex]
- Divide both sides by -1:
[tex]\[ y = -8 \][/tex]
So, the value of [tex]\( y \)[/tex] is [tex]\(-8\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{-8} \][/tex]
