To determine which equation is equivalent to [tex]\(4^{x+3}=64\)[/tex], let's break down the given equation step-by-step.
First, express 64 as a power of 4:
[tex]\[ 64 = 4^3 \][/tex]
So the equation [tex]\(4^{x+3} = 64\)[/tex] can be rewritten using this equality:
[tex]\[ 4^{x+3} = 4^3 \][/tex]
When bases are the same, we can set the exponents equal to each other:
[tex]\[ x + 3 = 3 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x + 3 - 3 = 3 - 3 \][/tex]
[tex]\[ x = 0 \][/tex]
This means we need to find the equation among the given options where the exponents equate similarly.
1. [tex]\(2^{x+6}=2^4\)[/tex]:
Converting [tex]\(x = 0\)[/tex] into the expression, this does not make sense because:
[tex]\[ 2^{0+6} \neq 2^4 \][/tex]
[tex]\[ 2^6 \neq 2^4 \][/tex]
2. [tex]\(2^{2x+6}=2^6\)[/tex]:
Converting [tex]\(x = 0\)[/tex] into the expression, we have:
[tex]\[ 2^{2(0)+6} = 2^6 \][/tex]
[tex]\[ 2^6 = 2^6 \][/tex]
This is true.
3. [tex]\(4^{2x+6}=4^2\)[/tex]:
Converting [tex]\(x = 0\)[/tex] into the expression, we have:
[tex]\[ 4^{2(0)+6} \neq 4^2 \][/tex]
[tex]\[ 4^6 \neq 4^2 \][/tex]
4. [tex]\(4^{x+3}=4^6\)[/tex]:
Converting [tex]\(x = 0\)[/tex] into the expression, this does not make sense because:
[tex]\[ 4^{0+3} \neq 4^6 \][/tex]
[tex]\[ 4^3 \neq 4^6 \][/tex]
Thus, the only correct option where the exponents match and the equations are equivalent is:
[tex]\[ 2^{2x+6} = 2^6 \][/tex]
Therefore, the correct equivalent equation is:
[tex]\[ 2^{2x+6}=2^6 \][/tex]
