Problem 19 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(z^{6}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified expression is \(z^{24}\).

Step 1: Identify the base and exponents

The given expression is \(\left(z^{6}\right)^{4}\). Here, the base is \(z\) and the exponents are \(6\) and \(4\).

Step 2: Apply the power rule for exponents

Using the power rule \((a^m)^n = a^{mn}\), we can rewrite the expression as follows: \((z^6)^4 = z^{6 \cdot 4}\)

Step 3: Multiply the exponents

Multiply the exponents, 6 and 4: \(6 \cdot 4 = 24\)

Step 4: Write the final answer

Replace the product of the exponents in the expression: \(z^{6 \cdot 4} = z^{24}\) The simplified expression without negative exponents is \(z^{24}\).

Key Concepts

Understanding the Power Rule

The power rule is a key concept in mathematics, especially when dealing with exponents. It simplifies expressions by allowing us to easily manage and reduce them. Here's a closer look:When you have an expression of the form \((a^m)^n\), the power rule says that you can multiply the exponents \(m\) and \(n\) together, resulting in \(a^{mn}\). This is incredibly useful for simplifying expressions with exponents. For example, in the expression \((z^6)^4\), the power rule tells us to multiply 6 and 4 together. So, we calculate \(6 \times 4 = 24\). Thus, \((z^6)^4\) simplifies to \(z^{24}\). This way, exponentiation becomes straightforward, reducing complex operations to simple multiplication tasks.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can represent real-world situations or abstract concepts in mathematics. Understanding how these expressions work is crucial for a variety of math problems.Variables in algebraic expressions, like the \(z\) in our example, stand for unknown or changeable values. They can have exponents, which indicate how many times the variable is used as a factor in multiplication. To simplify an algebraic expression with exponents, you often need to apply rules like the power rule. Remember, every part of an algebraic expression has a purpose and knowing what each part does is key to understanding and simplifying the expression effectively.

Simplifying Exponents

Simplifying exponents is about reducing expressions to their simplest form. This makes calculations easier and results clearer.With exponents, simplification usually involves applying certain rules, such as the power rule. This helps to combine and reduce the power of expressions to a single exponent, instead of multiple nested ones.For example, in \((z^6)^4\), simplifying involves recognizing that we can change the expression to something simpler by multiplying the exponents. We move from a double power situation to a single exponent \(z^{24}\). Simplification also involves ensuring there are no negative exponents, which might complicate the expression further. The end goal is a neat and simple presentation of the result, like making sure \(z^{24}\) does not have any hidden or unnecessary elements.