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Chapter 5: Problem 41

\(\left(2 z^{3}\right)^{3}(4 z)\)

Short Answer

Expert verified
32z^{10}

Step by step solution

01

Apply the Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{mn}\). Use this rule to simplify \( \left(2z^3 \right)^3\) by multiplying the exponents: \((2z^3)^3 = 2^3 \cdot (z^3)^3 = 8z^9\).
02

Combine with the Other Term

Now, multiply the simplified term \( 8z^9 \) by \( 4z \).
03

Multiply the Coefficients

Multiply the numerical coefficients together: \( 8 \times 4 = 32 \).
04

Add the Exponents

When multiplying like bases, add the exponents: \( z^9 \times z = z^{9+1} = z^{10} \).
05

Write the Final Answer

Combine the results from Steps 3 and 4 to get the final answer: \( 32z^{10} \).

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Key Concepts

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

Multiplying Exponents
In this section, let's dive into the concept of multiplying exponents. When you come across an expression like \( ( a^m )^n \), the Power of a Power Rule is your go-to rule. This states that you multiply the exponents to simplify the expression. For example, transforming \( ( 2z^3 )^3 \) involves multiplying the exponent 3 by itself: \( ( 2^1 z^3 )^3 = 2^3 \times (z^3)^3 = 8z^9 \). This is because you can treat the coefficient separately, raising it to the power of 3, while also raising the exponent on z to 3 times 3.
Combining Like Terms
Once you have simplified each part of the expression, you will often need to combine like terms. Like terms have the same variable raised to the same power. In the expression \( 8z^9 \times 4z \), combine the like terms by: \( 8 \times 4 \) for the coefficients, and then add the exponents for the z terms.
So, \( 8 \times 4 = 32 \) and \( z^9 \times z^1 = z^{9+1} = z^{10} \).
Simplifying Expressions
The final step involves putting together all you have worked out. After multiplying the coefficients and combining the exponents, you arrive at the simplified expression. For the original problem, \( \big(\big( 2z^3\big)^3\big) \times (4z) \), you get \( 32z^{10} \). Ensuring that all steps鈥攁pplying the Power of a Power Rule, combining terms, and simplifying fully鈥攁re done accurately is important for consistently getting the correct answers. Remember to keep equations neat and organized throughout the process to avoid mistakes.

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