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Chapter 10: Problem 6

Simplify. $$\left(-3 x^{2} z^{3}\right)^{2}$$

Short Answer

Expert verified
9x^4z^6

Step by step solution

01

Apply the Power Rule

When simplifying \(\big(-3x^2z^3\big)^2\), use the power rule \((a^m)^n = a^{mn}\). Apply the power to each factor inside the parentheses.
02

Simplify Each Element

First, apply the power of 2 to -3: \((-3)^2 = 9\). Next, apply the power to \(x^2\): \((x^2)^2 = x^{4}\). Finally, apply the power to \(z^3\): \((z^3)^2 = z^{6}\).
03

Combine the Results

Multiply all the simplified results together: \(( -3x^2z^3)^2 = 9x^4z^6\).

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Key Concepts

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

power rule
The power rule is a core concept you need to understand when simplifying exponents. It's a shortcut that helps you handle expressions where an exponent is raised to another exponent. The rule is expressed as \((a^m)^n = a^{mn}\). This means you multiply the exponents together. When you see \((x^2)^2\), you multiply the two exponents (2 * 2) to get \((x^4)\). It simplifies computations and keeps your work neat.
exponentiation
Exponentiation is the operation of raising one number to the power of another. In simpler terms, it's repeated multiplication of a base number. For example, \(-3^2 = (-3) \times (-3) = 9\). You deal with exponents when you encounter terms like \((x^2)^2\), \((z^3)^2\), etc. Remember, exponents help you manage large numbers without writing them out in full.
algebraic expressions
Algebraic expressions are collections of numbers, variables, and operations (addition, subtraction, multiplication, and division). When you encounter an expression like \(-3 x^2 z^3\), it means -3 is multiplied by \x^2\ and \z^3\. These expressions can be simplified using various algebraic rules and properties, making complex calculations easier and manageable.
multiplication of exponents
Multiplication of exponents follows a straightforward rule when you have the same base. For example, \((x^2) \times (x^3) = x^{2+3} = x^5\). When dealing with powers raised to other powers like \((x^2)^2\), use the power rule: multiply the exponents. Here, \((2 \times 2) = 4\), so \((x^2)^2 = x^4\). This rule makes it easy to handle and simplify complex expressions involving exponents.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{2}$$

Write a trinomial with degree 2.

Which is greater? \(\left(4^{4}\right)^{5}\) or \(\left(4^{5}\right)^{4}\)

Write the opposite of the polynomial. $$9 x^{2}-2 x-3$$

Simplify the expression. $$\left(y^{2}\right)^{3}\left(y^{4}\right)^{2}$$
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