Simplify each expression, but don’t evaluate.
${\left(4^{15}\right)}^3$

• A. $4^{18}$
• B. $4^5$
• C. $4^{12}$
• D. $4^{45}$

Answer: (D)

Verified step by step guidance

1. Identify the expression or problem involving exponents that you need to simplify or differentiate using the Power Rule. The Power Rule states that when you have a variable raised to a power, such as \(x^n\), the derivative is \(n \cdot x^{n-1}\).
2. If the problem involves simplifying expressions with exponents, recall the Power Rule for exponents: when multiplying like bases, add the exponents; when dividing like bases, subtract the exponents; and when raising a power to another power, multiply the exponents.
3. Apply the appropriate exponent rule based on the problem. For example, if you have \(x^a \cdot x^b\), rewrite it as \(x^{a+b}\). If you have \((x^a)^b\), rewrite it as \(x^{a \cdot b}\).
4. If the problem involves differentiation, apply the Power Rule by multiplying the exponent by the coefficient and then subtracting one from the exponent: \(\frac{d}{dx} x^n = n \cdot x^{n-1}\).
5. Simplify the resulting expression by combining like terms or rewriting the expression in simplest form, ensuring all exponents are correctly adjusted according to the Power Rule.