Let's solve the expression [tex]\(2^{7x + 6} \cdot 2^{3x - 4}\)[/tex] step-by-step using properties of exponents.
Recall the property of exponents that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In this case, our base [tex]\(a\)[/tex] is 2, and we have two exponents, [tex]\(7x + 6\)[/tex] and [tex]\(3x - 4\)[/tex].
First, let's identify the exponents in our expression:
[tex]\[ m = 7x + 6 \][/tex]
[tex]\[ n = 3x - 4 \][/tex]
According to the exponent rule:
[tex]\[ 2^{7x + 6} \cdot 2^{3x - 4} = 2^{(7x+6) + (3x-4)} \][/tex]
Next, we need to simplify the exponent [tex]\( (7x + 6) + (3x - 4) \)[/tex].
Combine the terms inside the parentheses:
[tex]\[ (7x + 6) + (3x - 4) = 7x + 3x + 6 - 4 \][/tex]
Now, add the coefficients of [tex]\(x\)[/tex] and the constants:
[tex]\[ 7x + 3x = 10x \][/tex]
[tex]\[ 6 - 4 = 2 \][/tex]
So, the simplified exponent is:
[tex]\[ 10x + 2 \][/tex]
Therefore, the expression [tex]\(2^{7x + 6} \cdot 2^{3x - 4}\)[/tex] simplifies to:
[tex]\[ 2^{10x + 2} \][/tex]
Thus, the simplified form of [tex]\(2^{7x + 6} \cdot 2^{3x - 4}\)[/tex] is [tex]\(2^{10x + 2}\)[/tex].
