Question

Simplify. $$ \left(4 c^{2}\right)^{4} $$

Short answer

The simplified form of \( (4c^{2})^{4} \) is \( 256c^{8} \)

Step-by-step solution

Rule recall

Recall the power of a power property, which states that multiplying the exponents together when a power is raised to a power. In the given problem, \( a = 4c^{2} \), \( m = 2 \), and \( n = 4 \)

Apply the rule

Applying the rule to \( (4c^{2})^{4} \), the exponents are multiplied together. For the \( c^{2} \) part, the exponent is multiplied by 4. For the 4 part, note that it can be written as \( 4^{1} \) so the rule is applied to it as well.

Simplify the exponents

4 to the power of 1 times 4 gives \( 4^{4} \), and \( c^{2*4} \) gives \( c^{8} \).

Final simplification

Calculate \( 4^{4} \) to get 256. So, the final answer after combining 256 and \( c^{8} \) is \( 256c^{8} \)

Key concepts

Power of a Power Property

Understanding the Power of a Power property is essential in simplifying algebraic expressions involving exponents. This property comes into play when an expression with an exponent is raised to another power, such as in the expression \(\left(a^{m}\right)^{n}\). To simplify, you multiply the exponents: \(m\times n\), resulting in \(a^{m\times n}\).For example, in the expression \(\left(4c^{2}\right)^{4}\), we apply this property. The constant 4 can be thought of as having an implicit exponent of 1, making it \(4^{1}\). When we raise it to the power of 4, we multiply the exponents:\[ 4^{1\times4} = 4^{4} \].Similarly, the variable component \(c^{2}\) raised to the power of 4 is handled by multiplying the exponents as well:\[ c^{2\times4} = c^{8} \].This property simplifies the process and requires us to focus on managing the exponents correctly.

Exponent Multiplication

Exponent multiplication is a key concept in algebra when you're dealing with exponential expressions. It refers to the process of multiplying the exponents when the same base is raised to different powers, emphasised by the power of a power property. This step is pivotal because it reduces complex expressions so that they can easily be calculated.In our example, we take the base 4 and raise it to power 1 and then to power 4. We also have the variable \(c\) with an exponent of 2, raised to the power of 4. Multiplying the exponents, we get:\[ 4^{1}\times4^{4} = 4^{1+4} = 4^{4} \]and\[ c^{2}\times c^{4} = c^{2+4} = c^{8} \].Through exponent multiplication, expressions become more straightforward to compute, breaking down a seemingly daunting task into an approachable calculation.

Algebraic Simplification

Algebraic simplification includes reducing expressions to their simplest form. It involves combining like terms, applying exponent rules, and performing arithmetic operations correctly. For the problem \(\left(4c^{2}\right)^{4}\), we've already applied the power of a power property and multiplied the exponents, but simplifying doesn't end there. We have \(4^{4}\) which is a numerical calculation, while \(c^{8}\) remains unchanged since it's a variable expression.To complete the simplification, compute \(4^{4}\), which is \(4\times4\times4\times4\), giving us 256. So, the expression simplifies to:\[ 256\times c^{8} \].Remember, algebraic simplification helps in making the expressions manageable, thus paving the way for further mathematical operations or evaluations.

Exponential Expressions

Exponential expressions are a form of mathematical notation where a number, called the base, is raised to a power, which is the exponent. For instance, \(4^{4}\) and \(c^{8}\) are both exponential expressions. The base tells us what number is being multiplied, and the exponent indicates how many times it is used as a factor.When dealing with exponential expressions such as these, it's important to recognize patterns and apply rules correctly for simplification. The expression \(4c^{2}\right)^{4}\) initially appears complex but understanding the associated rules like the power of a power property leads to a simplified exponential expression, in this case, \(256c^{8}\).Recognizing and simplifying exponential expressions are fundamental skills in algebra that assist in solving more advanced equations and in understanding the growth patterns described by exponents in real-world situations.