Answer :
To solve the equation [tex]\( \left(\frac{1}{4}\right)^{x+1} = 32 \)[/tex], we can take the following step-by-step approach:
1. Rewrite the equation in terms of exponents:
[tex]\[ \left(\frac{1}{4}\right)^{x+1} = 32 \][/tex]
2. Express the base [tex]\(\frac{1}{4}\)[/tex] as [tex]\((4)^{-1}\)[/tex]:
[tex]\[ \left(4^{-1}\right)^{x+1} = 32 \][/tex]
3. Simplify the left side using the properties of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ 4^{-(x+1)} = 32 \][/tex]
4. Rewrite 32 as a power of 2 for easier manipulation:
[tex]\[ 32 = 2^5 \][/tex]
5. Express the base 4 in terms of 2:
[tex]\[ 4 = 2^2 \][/tex]
Therefore,
[tex]\[ 4^{-(x+1)} = (2^2)^{-(x+1)} \][/tex]
Which simplifies to:
[tex]\[ 2^{-2(x+1)} = 2^5 \][/tex]
6. Since the bases are the same, set the exponents equal to each other:
[tex]\[ -2(x+1) = 5 \][/tex]
7. Solve for [tex]\(x\)[/tex]:
[tex]\[ -2(x+1) = 5 \][/tex]
[tex]\[ -2x - 2 = 5 \][/tex]
[tex]\[ -2x = 5 + 2 \][/tex]
[tex]\[ -2x = 7 \][/tex]
[tex]\[ x = -\frac{7}{2} \][/tex]
So, the solution to the equation [tex]\( \left(\frac{1}{4}\right)^{x+1} = 32 \)[/tex] is [tex]\( x = -\frac{7}{2} \)[/tex].
The correct answer is B. [tex]\( -\frac{7}{2} \)[/tex].
1. Rewrite the equation in terms of exponents:
[tex]\[ \left(\frac{1}{4}\right)^{x+1} = 32 \][/tex]
2. Express the base [tex]\(\frac{1}{4}\)[/tex] as [tex]\((4)^{-1}\)[/tex]:
[tex]\[ \left(4^{-1}\right)^{x+1} = 32 \][/tex]
3. Simplify the left side using the properties of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ 4^{-(x+1)} = 32 \][/tex]
4. Rewrite 32 as a power of 2 for easier manipulation:
[tex]\[ 32 = 2^5 \][/tex]
5. Express the base 4 in terms of 2:
[tex]\[ 4 = 2^2 \][/tex]
Therefore,
[tex]\[ 4^{-(x+1)} = (2^2)^{-(x+1)} \][/tex]
Which simplifies to:
[tex]\[ 2^{-2(x+1)} = 2^5 \][/tex]
6. Since the bases are the same, set the exponents equal to each other:
[tex]\[ -2(x+1) = 5 \][/tex]
7. Solve for [tex]\(x\)[/tex]:
[tex]\[ -2(x+1) = 5 \][/tex]
[tex]\[ -2x - 2 = 5 \][/tex]
[tex]\[ -2x = 5 + 2 \][/tex]
[tex]\[ -2x = 7 \][/tex]
[tex]\[ x = -\frac{7}{2} \][/tex]
So, the solution to the equation [tex]\( \left(\frac{1}{4}\right)^{x+1} = 32 \)[/tex] is [tex]\( x = -\frac{7}{2} \)[/tex].
The correct answer is B. [tex]\( -\frac{7}{2} \)[/tex].
