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Chapter 10: Problem 36

Simplify the expression. $$\left(9^{3}\right)^{8}$$

Short Answer

Expert verified
The expression simplifies to \(9^{24}\).

Step by step solution

01

Identify the expression

The given expression to simplify is \(\big(9^{3}\big)^{8}\).
02

Apply the Power Rule

Recall the power rule for exponents which states \(\big(a^{m}\big)^{n} = a^{m \times n}\). In this case, \a = 9\, \m = 3,\ and \ = 8.\
03

Multiply the Exponents

Multiply the exponents: \(3 \times 8 = 24\). Therefore, the expression simplifies to \9^{24}.

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Key Concepts

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

Power Rule for Exponents
The power rule for exponents is a fundamental concept in mathematics. It helps in simplifying expressions involving powers raised to another power. According to the rule, \(\big(a^{m}\big)^{n} = a^{m \times n}\). This means when you have an exponent raised to another exponent, you multiply the exponents together.

For example, if you have \( (9^3)^8 \), you apply the power rule to get \(9^{3 \times 8}\). The resulting expression, after performing the multiplication, will be \ 9^{24}\. This makes solving more manageable and quicker.
Multiplying Exponents
When using the power rule, one key step is multiplying the exponents. This step transforms complex expressions into simpler forms. If the base is the same in each term, this rule applies directly.

Consider \( (9^3)^8 \). To simplify this:
  • Identify the exponents (3 and 8).
  • Multiply the exponents: \( 3 \times 8 = 24 \).
After multiplying, the expression becomes \ 9^{24} \. This process reduces the expression to a single exponent, making it easier to understand and work with in further calculations.
Remember, this method only works correctly when the base number remains the same.
Simplification
Simplification in mathematics is about making expressions easier to handle and understand. Using properties like the power rule for exponents, complex expressions can be reduced to simpler forms. For example, the original expression \( (9^3)^8\) might look challenging initially. By applying the power rule and multiplying the exponents, we reduce it to a much simpler form: \(9^{24}\).

Here are some tips for simplification:
  • Identify any applicable rules or properties.
  • Apply these rules methodically.
  • Re-check your steps to ensure accuracy.
This ensures that the expression is as simplified as possible, avoiding mistakes and making further calculations more straightforward. Simplification is a crucial step in solving more complex mathematical problems efficiently.

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Most popular questions from this chapter

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

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