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Chapter 10: Problem 8

Simplify. $$\left(2 a^{5} b\right)^{4}$$

Short Answer

Expert verified
Simplified Expression: \16a^{20}b^4\.

Step by step solution

01

- Apply the Power Rule

When simplifying expressions with exponents, apply the power rule \( (x^m)^n = x^{mn} \). Here, each element inside the parentheses will be raised to the power outside the parentheses. So, apply \(4\) to each factor inside \((2 a^5 b)\). This leads to: \(2^4 (a^5)^4 b^4\).
02

- Simplify Each Factor

Now simplify each term. \(2^4 = 16\), \((a^5)^4 = a^{20}\), and \(b^4\). Combining these, the expression becomes \16a^{20}b^4\.

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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 fundamental concept in the realm of exponents. It states that when we raise an exponent to another exponent, we multiply the exponents together.
Mathematically, this is expressed as \( \left( x^m \right)^n = x^{mn} \).
In our original exercise, we used the power rule to simplify \( \left( 2a^5b \right)^4 \). Each element inside the parentheses is raised to the 4th power. This means:
  • \( 2^4 \)
  • \( \left( a^5 \right)^4 \)
  • \( b^4 \)
    • We apply the power rule separately to the base 2, to the term \( a^5 \), and to the base b.
    For instance, \( \left( a^5 \right)^4 = a^{5 \cdot 4} = a^{20} \).
    This is why it's essential to understand and use the power rule correctly!
Simplifying Expressions
Simplifying expressions involves breaking down complex expressions into their simpler forms.
This helps us solve them more easily and accurately.
In our exercise, the expression \( \left( 2a^5b \right)^4 \) was simplified by applying the power rule first, then working on each element separately.
After applying the power rule, our resulting expression was \( 2^4 \cdot ( a^{5 \cdot 4} ) \cdot b^4 \).
The next step in simplification was to simplify each term:
  • \( 2^4 = 16 \)
  • \( \left( a^5 \right)^4 = a^{20} \)
  • \( b^4 \)
    • Combining these simplified factors gives us the final result: \( 16a^{20}b^4 \).
    Simplifying exponents often requires us to handle several small steps, but doing so ensures clarity and correctness in our final answer.
Exponents
Exponents are a shorthand way to represent repeated multiplication of the same number or variable by itself.
For example, \( a^5 = a \cdot a \cdot a \cdot a \cdot a \).
They simplify writing and working with large numbers and variables.
Some key exponent rules include:
  • Product of Powers Rule: \( x^m \cdot x^n = x^{m+n} \)
  • Power of a Power Rule: \( \left( x^m \right)^n = x^{mn} \)
  • Power of a Product Rule: \( \left( xy \right)^n = x^n \cdot y^n \)
    • The third rule is particularly useful in our current exercise.
    We applied it when we initially tackled each element inside the parentheses \( \left( 2a^5b \right)^4 \).
    By raising each factor to the 4th power, we ultimately simplified our expression step-by-step and accurately.
    Understanding these rules allows us to manipulate and simplify expressions involving exponents with confidence.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$y^{-12} y^{14}$$

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