(Laws of Exponents with Integer Exponents)

Choose the expression that is equivalent to [tex]\left(7^{-2} \cdot 7^3\right)^{-3}[/tex].

A. [tex]-7^6[/tex]
B. [tex]\frac{1}{7^3}[/tex]
C. [tex]-\frac{1}{7^6}[/tex]
D. [tex]7^{18}[/tex]



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]