Alright, let’s solve the given equations step by step.
### Part d
We are given the equation:
[tex]\[ 58^8 \cdot 58^y = 58^{55} \][/tex]
To solve for [tex]\( y \)[/tex], we can use the property of exponents which states that [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]. This allows us to combine the exponents of the same base.
Using this property, we get:
[tex]\[ 58^{8 + y} = 58^{55} \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 8 + y = 55 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = 55 - 8 \][/tex]
[tex]\[ y = 47 \][/tex]
So, the value of [tex]\( y \)[/tex] in part d is [tex]\( \boxed{47} \)[/tex].
### Part e
We are given the equation:
[tex]\[ \left(\frac{3}{4}\right)^y = 1 \][/tex]
To solve for [tex]\( y \)[/tex], we need to recall the property that any non-zero number raised to the power of zero is 1. This implies that:
[tex]\[ \left(\frac{3}{4}\right)^0 = 1 \][/tex]
Thus, the only exponent that makes the expression [tex]\( \left(\frac{3}{4}\right)^y \)[/tex] equal to 1 is:
[tex]\[ y = 0 \][/tex]
So, the value of [tex]\( y \)[/tex] in part e is [tex]\( \boxed{0} \)[/tex].
Therefore, the solutions to the equations are:
- For part d: [tex]\( y = 47 \)[/tex]
- For part e: [tex]\( y = 0 \)[/tex]
