(v) $$ \left(5^{4}\right)^{2}=5^{8} $$

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UpStudy AI · Accepted Answer

Yes, your step (v) is correct. Here's a detailed explanation:

### Given Expression:
$$
\left(5^{4}\right)^{2}
$$

### Applying the Power of a Power Rule:
When you raise a power to another power, you multiply the exponents. This is known as the **Power of a Power Rule** in exponents, which states:
$$
\left(a^{m}\right)^{n} = a^{m \times n}
$$

### Applying the Rule to Your Expression:
$$
\left(5^{4}\right)^{2} = 5^{4 \times 2} = 5^{8}
$$

### Final Simplified Form:
$$
5^{8}
$$

So, indeed:
$$
\left(5^{4}\right)^{2} = 5^{8}
$$

### Additional Context (Optional):

If you're working on simplifying exponential expressions or solving equations involving exponents, understanding and correctly applying the exponent rules is crucial. Here's a brief overview of some key exponent rules that might be helpful:

1. **Product of Powers Rule**:
   $$
   a^{m} \times a^{n} = a^{m + n}
   $$
   
2. **Quotient of Powers Rule**:
   $$
   \frac{a^{m}}{a^{n}} = a^{m - n}
   $$
   
3. **Power of a Power Rule**:
   $$
   \left(a^{m}\right)^{n} = a^{m \times n}
   $$
   
4. **Power of a Product Rule**:
   $$
   (ab)^{n} = a^{n} \times b^{n}
   $$
   
5. **Zero Exponent Rule**:
   $$
   a^{0} = 1 \quad \text{(where } a \neq 0\text{)}
   $$
   
6. **Negative Exponent Rule**:
   $$
   a^{-n} = \frac{1}{a^{n}} \quad \text{(where } a \neq 0\text{)}
   $$

Understanding these rules will help you manipulate and simplify exponential expressions effectively.
$$ \left(5^{4}\right)^{2} = 5^{8} $$

(v)

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