Answer :
To solve the expression \(\left(7^{-2} \cdot 7^3\right)^{-3}\) step-by-step, let's use the laws of exponents.
### Step 1: Simplify inside the parentheses
First, we have the expression inside the parentheses: \(7^{-2} \cdot 7^3\).
Using the product of powers property \(a^m \cdot a^n = a^{m+n}\), we combine the exponents:
[tex]\[7^{-2} \cdot 7^3 = 7^{-2 + 3}\][/tex]
### Step 2: Combine the exponents
Calculate the exponent inside the parentheses:
[tex]\[-2 + 3 = 1\][/tex]
So, the expression simplifies to:
[tex]\[7^1\][/tex]
Thus, the expression becomes:
[tex]\[\left(7^1\right)^{-3}\][/tex]
### Step 3: Apply the power of a power property
Now we have \(\left(7^1\right)^{-3}\). To simplify this, we use the power of a power property \((a^m)^n = a^{m \cdot n}\):
[tex]\[7^{1 \cdot (-3)}\][/tex]
### Step 4: Multiply the exponents
Multiply the exponents:
[tex]\[1 \cdot (-3) = -3\][/tex]
So, the expression simplifies to:
[tex]\[7^{-3}\][/tex]
### Step 5: Write as a fraction
By definition, a negative exponent indicates the reciprocal:
[tex]\[7^{-3} = \frac{1}{7^3}\][/tex]
Hence, the expression \(\left(7^{-2} \cdot 7^3\right)^{-3}\) is equivalent to \(\frac{1}{7^3}\).
### Conclusion
The equivalent expression is:
[tex]\[ \boxed{\frac{1}{7^3}} \][/tex]
### Step 1: Simplify inside the parentheses
First, we have the expression inside the parentheses: \(7^{-2} \cdot 7^3\).
Using the product of powers property \(a^m \cdot a^n = a^{m+n}\), we combine the exponents:
[tex]\[7^{-2} \cdot 7^3 = 7^{-2 + 3}\][/tex]
### Step 2: Combine the exponents
Calculate the exponent inside the parentheses:
[tex]\[-2 + 3 = 1\][/tex]
So, the expression simplifies to:
[tex]\[7^1\][/tex]
Thus, the expression becomes:
[tex]\[\left(7^1\right)^{-3}\][/tex]
### Step 3: Apply the power of a power property
Now we have \(\left(7^1\right)^{-3}\). To simplify this, we use the power of a power property \((a^m)^n = a^{m \cdot n}\):
[tex]\[7^{1 \cdot (-3)}\][/tex]
### Step 4: Multiply the exponents
Multiply the exponents:
[tex]\[1 \cdot (-3) = -3\][/tex]
So, the expression simplifies to:
[tex]\[7^{-3}\][/tex]
### Step 5: Write as a fraction
By definition, a negative exponent indicates the reciprocal:
[tex]\[7^{-3} = \frac{1}{7^3}\][/tex]
Hence, the expression \(\left(7^{-2} \cdot 7^3\right)^{-3}\) is equivalent to \(\frac{1}{7^3}\).
### Conclusion
The equivalent expression is:
[tex]\[ \boxed{\frac{1}{7^3}} \][/tex]
