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Chapter 5: Problem 103

Simplify each expression, if possible. $$ \left(-6 a^{3} b^{2}\right)^{3} $$

Short Answer

Expert verified
The simplification of \((-6 a^{3} b^{2})^{3}\) is \(-216a^9b^6\).

Step by step solution

01

Apply the Power of a Power Rule

When raising an expression with exponents to another power, multiply the exponents for each variable. The power of a power rule states: \[(a^m)^n = a^{m \cdot n}\]Apply this rule to each term inside the parentheses:\[(-6 a^{3} b^{2})^3 = (-6)^3 (a^{3})^{3} (b^{2})^{3}\]
02

Simplify Each Term

Now simplify each part of the expression:1. For \((-6)^3\), calculate the cube of -6: \[(-6)^3 = -6 \times -6 \times -6 = -216\]2. Apply the power of a power rule to the variables: \[(a^{3})^{3} = a^{3 \cdot 3} = a^9\] \[(b^{2})^{3} = b^{2 \cdot 3} = b^6\]
03

Combine Simplified Terms

Combine all the simplified terms into a single expression:\[(-216) \cdot a^9 \cdot b^6\]The final expression is:\[-216a^9b^6\]

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Key Concepts

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

The Power Rule in Exponents
The power rule in exponents is a handy tool when dealing with expressions that have exponents raised to another power. This rule helps you simplify these expressions easily. The power rule states that when you have
  • an exponent raised to another exponent, like \( \left( a^m \right)^n \), you can multiply the exponents together to simplify the expression.
  • For example, this would become \( a^{m \cdot n} \).
Imagine you have \( a^3 \) and you raise it to the power of 3 again:
  • \( \left( a^3 \right)^3 \) would become \( a^{3 \times 3} \), which simplifies to \( a^9 \).
This concept is crucial because it allows you to reduce complex-looking expressions into a more manageable format, making calculations easier and less prone to errors. The power rule is your go-to strategy whenever you see a power raised to another power in an algebraic expression.
Simplifying Expressions
Simplifying expressions involves breaking down complex expressions into simpler, more understandable forms. This is essential for solving algebraic equations efficiently. To simplify an expression like \( \left(-6 a^3 b^2 \right)^3 \), follow these steps:
  • First, apply the power rule as explained before.
  • Next, handle each part within the parentheses separately. Raise \(-6\), \(a^3\), and \(b^2\) to the power of 3.
  • Cubing -6 involves calculating \(-6 \times -6 \times -6\), resulting in \(-216\).
  • Then, simplify \(a^3\) raised to the power 3 by multiplying exponents: \(3 \times 3 = 9\), so it becomes \(a^9\).
  • Similarly, simplify \(b^2\) raised to the power 3: \(2 \times 3 = 6\), making it \(b^6\).
The final step is combining all your simplified parts to get the neat expression \(-216a^9b^6\). This simplified form is much cleaner and clearer for further use in calculations.
Understanding Algebraic Expressions
Algebraic expressions are a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. Understanding expressions like \( \left(-6 a^3 b^2 \right)^3 \) involves recognizing the role of each component:
  • **Numbers or coefficients** - these are numerical values that multiply the variables. In our case, \(-6\) is the coefficient.
  • **Variables** - symbols, like \(a\) and \(b\), that represent numbers. Variables enable you to create equations that can solve a wide range of problems beyond fixed numbers. Here, \(a^3\) and \(b^2\) show how variables can be raised to powers.
  • **Operations** - including powers, addition, and subtraction, dictate how to manipulate these numbers and variables.
Breaking down expressions into parts helps you understand how different elements work together. By visualizing these components, you can apply rules like the power rule effectively and simplify expressions accurately. This foundational understanding is crucial in advancing in algebra and other areas of mathematics.

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Most popular questions from this chapter

Explain the error: \((x+3)(x-2)=x^{2}-6\)

Perform each division $$ \frac{6 a^{3}-17 a^{2} b+14 a b^{2}-3 b^{3}}{2 a-3 b} $$

Perform the indicated operations to simplify each expression, if possible. $$ \text { a. }(x-2)+\left(x^{2}+2 x+4\right) \text { b. }(x-2)\left(x^{2}+2 x+4\right) $$

Perform each division. Divide \(z^{2}+7 z+14\) by \(z+3\)

When dividing \(x^{3}+1\) by \(x+1,\) why is it helpful to write \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1 ?\)
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