S Which expressions are equivalent to $$ 24^{-7} $$ ? Choose all the correct answers. A $$ \frac{24^{-3}}{24^{-4}} $$ B $$ 3^{-2} \cdot 8^{-5} $$ C $$ \left(\frac{1}{24^{-7}}\right)^{-1} $$ D $$ \frac{8^{-7}}{3^{7}} $$ (E) $$ 24^{7} \cdot 24^{0} $$ F $$ \left(24^{3} \cdot 24^{4}\right)-1 $$

Question

UpStudy AI · Accepted Answer

To determine which expressions are equivalent to $$ 24^{-7} $$, we will simplify each option step by step.

### Option A: $$ \frac{24^{-3}}{24^{-4}} $$
Using the property of exponents $$ \frac{a^m}{a^n} = a^{m-n} $$:
$$
\frac{24^{-3}}{24^{-4}} = 24^{-3 - (-4)} = 24^{-3 + 4} = 24^{1}
$$
This is not equivalent to $$ 24^{-7} $$.

### Option B: $$ 3^{-2} \cdot 8^{-5} $$
We need to express $$ 8^{-5} $$ in terms of base $$ 24 $$:
$$
8 = 2^3 \implies 8^{-5} = (2^3)^{-5} = 2^{-15}
$$
Now, we can express $$ 3^{-2} $$ as is:
$$
3^{-2} = \frac{1}{3^2}
$$
Thus, $$ 3^{-2} \cdot 8^{-5} = \frac{1}{3^2} \cdot 2^{-15} $$, which does not relate to $$ 24^{-7} $$ (since $$ 24 = 2^3 \cdot 3^1 $$). Therefore, this is not equivalent.

### Option C: $$ \left(\frac{1}{24^{-7}}\right)^{-1} $$
Using the property $$ \left(\frac{1}{a}\right)^{-1} = a $$:
$$
\left(\frac{1}{24^{-7}}\right)^{-1} = 24^{-7}
$$
This is equivalent to $$ 24^{-7} $$.

### Option D: $$ \frac{8^{-7}}{3^{7}} $$
We can express $$ 8^{-7} $$ in terms of base $$ 24 $$:
$$
8 = 2^3 \implies 8^{-7} = (2^3)^{-7} = 2^{-21}
$$
Thus, we have:
$$
\frac{8^{-7}}{3^{7}} = \frac{2^{-21}}{3^{7}}
$$
This does not relate to $$ 24^{-7} $$ (since $$ 24 = 2^3 \cdot 3^1 $$). Therefore, this is not equivalent.

### Option E: $$ 24^{7} \cdot 24^{0} $$
Using the property $$ a^m \cdot a^n = a^{m+n} $$:
$$
24^{7} \cdot 24^{0} = 24^{7 + 0} = 24^{7}
$$
This is not equivalent to $$ 24^{-7} $$.

### Option F: $$ \left(24^{3} \cdot 24^{4}\right)-1 $$
Using the property $$ a^m \cdot a^n = a^{m+n} $$:
$$
24^{3} \cdot 24^{4} = 24^{3 + 4} = 24^{7}
$$
Thus, we have:
$$
\left(24^{3} \cdot 24^{4}\right) - 1 = 24^{7} - 1
$$
This is not equivalent to $$ 24^{-7} $$.

### Summary of Results
The only expression that is equivalent to $$ 24^{-7} $$ is:
- **C** $$ \left(\frac{1}{24^{-7}}\right)^{-1} $$

Thus, the correct answer is **C**.
The correct answer is **C** $$ \left(\frac{1}{24^{-7}}\right)^{-1} $$.

S Which expressions are equivalent to ? Choose all the correct answers.
A
B
C
D
(E)
F

Upstudy AI Solution

Answer

Solution

Option A:

Option B:

Option C:

Option D:

Option E:

Option F:

Summary of Results

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