Certainly! Let's solve the given problems step by step.
Step 1: Solve for [tex]\( x \)[/tex] when [tex]\( 2^x = 64 \)[/tex]
We need to determine the value of [tex]\( x \)[/tex] in the equation [tex]\( 2^x = 64 \)[/tex].
We know that 64 is a power of 2:
[tex]\[ 64 = 2^6 \][/tex]
So, equating the exponents we get:
[tex]\[ x = 6 \][/tex]
Step 2: Calculate [tex]\( 3^x \)[/tex]
Now that we know [tex]\( x = 6 \)[/tex], we can calculate [tex]\( 3^x \)[/tex]:
[tex]\[ 3^x = 3^6 \][/tex]
The value of [tex]\( 3^6 \)[/tex] is:
[tex]\[ 3^6 = 729 \][/tex]
So,
[tex]\[ \text{a. } 3^x = 729 \][/tex]
Step 3: Calculate [tex]\( 2^{x-3} \)[/tex]
Next, we need to find [tex]\( 2^{x-3} \)[/tex]. Using [tex]\( x = 6 \)[/tex]:
[tex]\[ 2^{x-3} = 2^{6-3} = 2^3 \][/tex]
The value of [tex]\( 2^3 \)[/tex] is:
[tex]\[ 2^3 = 8 \][/tex]
So,
[tex]\[ \text{b. } 2^{x-3} = 8 \][/tex]
Step 4: Calculate [tex]\( 4^{2x-3} \)[/tex]
Finally, we need to find [tex]\( 4^{2x-3} \)[/tex]. Substituting [tex]\( x = 6 \)[/tex]:
[tex]\[ 4^{2x-3} = 4^{2 \cdot 6 - 3} = 4^{12 - 3} = 4^9 \][/tex]
The value of [tex]\( 4^9 \)[/tex] is:
[tex]\[ 4^9 = (2^2)^9 = 2^{18} \][/tex]
[tex]\[ 2^{18} = 262144 \][/tex]
So,
[tex]\[ \text{c. } 4^{2x-3} = 262144 \][/tex]
Summary:
- [tex]\( x = 6 \)[/tex]
- [tex]\( 3^x = 729 \)[/tex]
- [tex]\( 2^{x-3} = 8 \)[/tex]
- [tex]\( 4^{2x-3} = 262144 \)[/tex]
That concludes our calculations for the given problem.
