Complete the Expression: Finding (15)^xy Power with Multiple Variables

To solve this problem, we will rewrite the expression $(15)^{xy}$ using the rules of exponents.

• Step 1: Understand that $(15)^{xy}$ can be rewritten using the power of a power rule.
• Step 2: Apply the exponent rule $(a^m)^n = a^{m \times n}$. We know $(15^x)^y = (15)^{x \times y}$ and $(15^y)^x = (15)^{y \times x}$, both equivalent to $(15)^{xy}$.
• Step 3: Analyze each choice:

Choice 1: $(15^y)^x$ is equivalent to $(15)^{xy}$ since applying the rule gives us $(15^y)^x = (15)^{y \times x} = (15)^{xy}$.
Choice 2: $(15^x)^y$ is also equivalent to $(15)^{xy}$ because applying the rule provides $(15^x)^y = (15)^{x \times y} = (15)^{xy}$.
Choice 3: $15^x \times 15^y$ results in $15^{x+y}$, which is not equivalent to $(15)^{xy}$ as it uses the product of powers rule.\
Choice 4: Both $(15^y)^x$ and $(15^x)^y$ are correct based on the rules involved.

Based on the analysis, choice 4 (a'+b' are correct) is the correct answer.
Both $(15^y)^x$ and $(15^x)^y$ are equivalent representations of $(15)^{xy}$.