Question

1. Use the Power Rule of Exponents to simplify the expression (6^2)^4

2. Use the Power Rule of Exponents to simplify the expression (73^4)^5

3. Use the Power Rule of Exponents to simplify the expression (78^8)^6

4. Use the Power Rule of Exponents to create a simplified equivalent expression of (12 x 12 x 12) x (12x12x12)

5. What exponent would make the two expressions equivalent: (16^x)^6 = (16^18)

Answer 1

Let's simplify each expression using the Power Rule of Exponents, which states that \((a^m)^n = a^{m \cdot n}\).

1. For the expression \((6^2)^4\):
\[
(6^2)^4 = 6^{2 \cdot 4} = 6^8
\]

2. For the expression \((73^4)^5\):
\[
(73^4)^5 = 73^{4 \cdot 5} = 73^{20}
\]

3. For the expression \((78^8)^6\):
\[
(78^8)^6 = 78^{8 \cdot 6} = 78^{48}
\]

4. To simplify the expression \((12 \times 12 \times 12) \times (12 \times 12 \times 12)\), we can count the total number of 12's:
\[
(12^3) \times (12^3) = 12^{3 + 3} = 12^6
\]

5. For the equation \((16^x)^6 = (16^{18})\), we use the Power Rule:
\[
16^{x \cdot 6} = 16^{18}
\]
This implies:
\[
x \cdot 6 = 18
\]
Now, solve for \(x\):
\[
x = \frac{18}{6} = 3
\]

In summary:
1. \(6^8\)
2. \(73^{20}\)
3. \(78^{48}\)
4. \(12^6\)
5. \(x = 3\)

Answer 2

Which expression is correctly developed to use the Power Rule of Exponents?
(7^4)^8
12^9
(7^8) x (7^10)
(-5)^0

What exponent would make the two expressions equivalent: (3^3)^x = 3^24

Answer 3

Let's analyze each expression to determine if they are correctly developed to use the Power Rule of Exponents.

1. **Expression**: \((7^4)^8\)
- This expression can be simplified using the Power Rule, as follows:
\((7^4)^8 = 7^{4 \cdot 8} = 7^{32}\)
- **Correctly developed**.

2. **Expression**: \(12^9\)
- This expression does not apply the Power Rule, as it is already expressed in simplified form.
- **Not applicable** for the Power Rule.

3. **Expression**: \((7^8) \times (7^{10})\)
- This expression combines two bases with the same base, which can be simplified as follows:
\(7^8 \times 7^{10} = 7^{8 + 10} = 7^{18}\)
- **Correctly developed**.

4. **Expression**: \((-5)^0\)
- By the Zero Exponent Rule, any non-zero number raised to the power of zero equals 1. This does not use the Power Rule but rather applies the zero exponent property.
- **Not applicable** for the Power Rule.

### Summary of Correctly Developed Expressions:
- \((7^4)^8\) is correctly developed.
- \((7^8) \times (7^{10})\) is correctly developed.
- \(12^9\) and \((-5)^0\) do not apply the Power Rule in any transformation.

Now, for the given exponential equation:

**To solve**: \((3^3)^x = 3^{24}\)

Using the Power Rule:
\[
(3^3)^x = 3^{3x}
\]
Set the exponents equal to each other:
\[
3x = 24
\]
Now, divide both sides by 3:
\[
x = \frac{24}{3} = 8
\]

### The exponent that makes the two expressions equivalent is:
\(x = 8\)

Answer 4

1. (7^4)^8

2. 10^63

3. (4^5)^2

4. 15^12

5. 8

Answer 5

Let's analyze each expression in relation to the Power Rule of Exponents, which states that \((a^m)^n = a^{m \cdot n}\).

### 1. \((7^4)^8\)
Using the Power Rule:
\[
(7^4)^8 = 7^{4 \cdot 8} = 7^{32}
\]
- This is **correctly developed**.

### 2. \(10^{63}\)
This expression does not apply the Power Rule directly, as it is already in its simplest form.
- Not applicable for the Power Rule but is **correct** in its current form.

### 3. \((4^5)^2\)
Using the Power Rule:
\[
(4^5)^2 = 4^{5 \cdot 2} = 4^{10}
\]
- This is **correctly developed**.

### 4. \(15^{12}\)
Similar to \(10^{63}\), this expression is in its simplest form and does not apply the Power Rule.
- Not applicable for the Power Rule but is **correct** as it stands.

### 5. \(8\)
This expression does not make use of exponents in any form. However, if intended, it can be expressed as \(8^1\) (which does not apply the Power Rule).
- Not applicable for the Power Rule.

### Summary:
The expressions that correctly use the Power Rule of Exponents are:
- \((7^4)^8\) simplifies to \(7^{32}\).
- \((4^5)^2\) simplifies to \(4^{10}\).
- Both \(10^{63}\) and \(15^{12}\) are already defined correctly but do not illustrate the Power Rule since they are not products of exponents that need simplification.

Overall, the final results from the original equations are:
1. \(7^{32}\)
2. \(10^{63}\) (no simplification needed)
3. \(4^{10}\)
4. \(15^{12}\) (no simplification needed)
5. \(8\) (but can be expressed as \(8^1\) if necessary).