Let's evaluate the expression step by step to find the values of the variables \( a \), \( b \), \( c \), and \( d \).
Given expression:
\[ \frac{45(-2)^9}{48(-2)^3} \]
Step 1: Apply the quotient of powers
Using the properties of exponents, specifically \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the powers of \((-2)\):
\[ (-2)^9 \text{ and } (-2)^3 \rightarrow \frac{(-2)^9}{(-2)^3} = (-2)^{9-3} = (-2)^6 \]
So after applying the quotient of powers, we have:
\[ \frac{45}{48} \cdot (-2)^{6} \]
From here, we can assign our variables:
- \( a = 6 \) (the exponent on the \(-2\))
Step 2: Evaluate powers
Now we need to find the values of \( c \) and \( d \). First, let's evaluate the fraction:
\[ \frac{45}{48} \] We can simplify this fraction. The greatest common divisor (GCD) of 45 and 48 is 3:
\[ \frac{45 \div 3}{48 \div 3} = \frac{15}{16} \]
Now we have the expression:
\[ \frac{15}{16} \cdot (-2)^6 \]
Next, we calculate \((-2)^6\):
\[ (-2)^6 = 64 \]
Now, we can evaluate the entire expression:
\[ \frac{15}{16} \cdot 64 = \frac{15 \cdot 64}{16} = \frac{960}{16} = 60 \]
Thus, for \( c \) and \( d \):
- \( c = 60 \) (final evaluated expression)
- \( d = 16 \) (the denominator of the simplified fraction)
Final Values
Putting it all together, we have:
- \( a = 6 \)
- \( b = 16 \)
- \( c = 60 \)
- \( d = 16 \)
So:
\[ \begin{align*} a & = 6 \ b & = 16 \ c & = 60 \ d & = 16 \ \end{align*} \]