Question

Write your answer as a power or as a product of powers. $$ \left[(-4)^{2}\right]^{3} $$

Short answer

The simplified form of \(\left[(-4)^{2}\right]^{3}\) is \((-4)^6\).

Step-by-step solution

Write down the expression

The expression given in the exercise is \(\left[(-4)^{2}\right]^{3}\). We have to simplify it.

Apply the rules of exponentiation

According to the rules of exponentiation, when raising a power to a power, you multiply the exponents. Therefore, we will multiply 2 (the exponent of -4) by 3 (the outer exponent). This gives us a new exponent for -4, which is 2*3 = 6.

Write the final answer

After applying the rule of exponentiation, the simplified form of \(\left[(-4)^{2}\right]^{3}\) is \((-4)^6\).

Key concepts

Raising a Power to a Power

Understanding how to handle an expression where a number with an exponent is raised to another power is a fundamental skill in algebra. This operation is often referred to as raising a power to a power. The rule is straightforward: when you have an expression in the form of \( (a^n)^m \), you multiply the exponents together to simplify the expression. The result is \( a^{n*m} \).

For example, let's examine the expression given in the exercise, \(\left[(-4)^{2}\right]^{3}\). Here, we raise \( -4 \) squared to the third power. Using our rule, we multiply the exponents 2 and 3 to find our new exponent, resulting in \( -4^6 \).

Why This Works

Raising a power to another power means you're multiplying the base number by itself several times. The count of these multiplications is equal to the product of the two exponents. This concept comes from the properties of exponents, which are a foundational piece of algebra.

Simplifying Expressions

The art of simplifying expressions in algebra involves reducing an expression to its simplest form while keeping its value unchanged. To simplify an expression with exponents, we consolidate the powers by using the rules of exponentiation.

It's worth noting that simplification makes complex algebraic expressions more manageable and easier to understand. For instance, simplifying \(\left[(-4)^{2}\right]^{3}\) doesn't just mean using the raising a power to a power rule. It also means understanding the significance of the negative base. The base \( -4 \) is within parentheses, meaning the negative sign is part of the base raised to the power of 2 first, before being raised to the third power. This simplifies to \( (-4)^6 \), which confirms that \( -4 \) is used as the base for six multiplications.

Algebraic Operations

Algebra involves a variety of algebraic operations, including addition, subtraction, multiplication, division, and exponentiation. Each operation follows specific rules that must be understood to solve equations and simplify expressions properly. In the context of exponentiation, we focus on the multiplication of bases and the handling of their exponents.

For example, when an expression requires raising a number to a power, as in the provided exercise, we are dealing with an operation where the base stays the same, but the effect of the exponents is combined through multiplication, another fundamental algebraic operation. Therefore, mastering the operation of raising a power to a power is not an isolated skill; it's part of the broader set of algebraic tools that allow for the simplification and solution of algebraic expressions.