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Chapter 5: Problem 64

\(\left(-z^{7}\right)^{3}\)

Short Answer

Expert verified
\(-z^{21}\)

Step by step solution

01

- Understand the Power Rule

Recall that \(\big(a^m\big)^n = a^{m \times n}\). This means you can multiply the exponents when raising a power to another power.
02

- Apply the Power Rule

Given the expression \(\big(-z^7\big)^3\), apply the power rule: \(\big(-z^7\big)^3 = -z^{7 \times 3}\).
03

- Simplify the Exponent

Multiply the exponents: \(-z^{7 \times 3} = -z^{21}\).

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Key Concepts

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

Power Rule in Exponentiation
The power rule in exponentiation is a fundamental concept in algebra. It simplifies expressions where an exponent is raised to another exponent.
For example, if you have an expression like \(\big(a^m\big)^n\), the power rule tells us to multiply the exponents together. This gives us \(a^{m \times n}\).
This rule helps to condense and simplify expressions, making them easier to work with.
Let's look at the exercise \(\big(-z^7\big)^3\). According to the power rule, you multiply 7 and 3 together to get \(-z^{21}\).
This way, you have turned a more complex expression into a simpler one.
Multiplying Exponents
Multiplying exponents is a key skill when working with powers and roots.
In the previous example, you encountered two exponents that needed to be multiplied. The general rule is: if you have \(\big(a^m\big)^n\), multiply the exponents \(m\) and \(n\).
Here's how to do it:
First, identify the base (in our example it's \(-z\)). Next, identify the exponents (7 and 3). Finally, multiply the exponents: 7 times 3 equaling 21.
So, \(\big(-z^7\big)^3\) becomes \(-z^{21}\).
Always make sure you are careful with negative bases. In this problem, -z raised to any power will keep z in the base negative, simplifying the work.
Simplifying Expressions
Simplifying expressions is crucial for clear, concise mathematics. Simplification helps to reduce complicated-looking terms into more manageable ones.
Starting from \(\big(-z^7\big)^3\), we have simplified it by using exponent rules. First, we applied the power rule, which gave us \(\big(-z^7\big)^3 = -z^{7 \times 3}\).
Next, by multiplying the exponents, we further simplified the expression to \(-z^{21}\).
Remember, the main goal in algebra is to make expressions as simple as possible. Look for opportunities to apply rules like the power rule, and make sure you perform operations like multiplying exponents carefully for accurate simplifications.

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