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Chapter 9: Problem 50

Simplify. $$\left(z^{4}\right)^{3}$$

Short Answer

Expert verified
The simplified form of \( (z^{4})^{3} \) is \( z^{12} \)

Step by step solution

01

Identify the base and the exponents

The base is 'z' while the exponents are \(4\) and \(3\). The expression \( (z^{4})^{3} \) means 'z' raised to the power \(4\), and then this result is raised to the power \(3\).
02

Apply the rule of exponents

According to the rules of exponents, when an exponent is raised to another exponent, we multiply the exponents. So, \( (z^{4})^{3} = z^{4*3} \)
03

Simplify the exponent

Multiplying the exponents, we get \( z^{12} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions involves making them easier to understand and work with. When dealing with powers and exponents, simplification often means combining like terms or applying specific rules to reduce expressions.
In our example, we have the expression \((z^{4})^{3}\). Here, simplification means applying rules to transform and reduce the expression to a simpler form. The goal is to express it in the simplest way possible, which in this case, means performing the multiplication within the exponents and rewriting it as \(z^{12}\). This process helps make equations less complex and more efficient to solve.
Rules of Exponents
The rules of exponents are crucial for simplifying expressions with powers. They provide guidelines on how to manipulate expressions involving exponents.
Some common rules include:
  • Product Rule: When multiplying like bases, add their exponents: \(a^{m} \cdot a^{n} = a^{m+n}\).
  • Quotient Rule: When dividing like bases, subtract their exponents: \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
  • Power of a Power Rule: When raising an exponent to another exponent, multiply the exponents: \((a^{m})^{n} = a^{m \cdot n}\).
In our expression \((z^{4})^{3}\), we specifically use the power of a power rule to simplify it. Understanding these rules helps in efficiently solving and simplifying expressions in mathematics.
Power of a Power Rule
The power of a power rule is an essential rule that simplifies expressions with exponents. When you have an exponent raised to another exponent, this rule tells you to multiply the exponents together.
For instance, in the expression \((z^{4})^{3}\), the base \(z\) is first raised to the power of 4, and then this result is raised to the power of 3. According to the power of a power rule, you multiply these exponents: \(4 \cdot 3 = 12\).
This transforms the expression to \(z^{12}\). Applying this rule makes calculations quicker and is a powerful tool when dealing with complex expressions involving multiple layers of exponents.

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