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Chapter 7: Problem 8

Compute the value of each of the following exponential expressions. $$ -3(-3)^{3} $$

Short Answer

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Step by step solution

01

Understand the Given Expression

The given expression \( \left(3^{2}\right)^{2} \) is a power of a power. This can be simplified using the properties of exponents.
02

Apply the Power of a Power Property

Use the rule \( \left(a^{m}\right)^{n} = a^{m \cdot n} \) to simplify the expression. Here, \( a = 3 \), \( m = 2 \), and \( n = 2 \).
03

Simplify the Exponents

Multiply the exponents: \( 3^{2 \cdot 2} = 3^{4} \).
04

Calculate the Final Value

Evaluate \( 3^{4} \). Since \( 3^{4} = 3 \times 3 \times 3 \times 3 = 81 \), the final value is 81.

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Key Concepts

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

Power of a Power Property
The 'Power of a Power' property is a key concept in exponential expressions. It helps in simplifying expressions where an exponent is raised to another exponent. The rule for this property is: \[ \left(a^{m}\right)^{n} = a^{m \cdot n} \]
Here鈥檚 a quick breakdown:
  • \
Simplifying Expressions
Simplifying exponential expressions can make them easier to work with and understand. Using the properties of exponents is crucial in this process.
We'll break down the expression step-by-step:
  • Identify the properties of exponents relevant to the expression.
  • Simplify the expression one property at a time.

For example, \[ \left(3^2\right)^2 \] simplifies as follows:
  • Apply the 'Power of a Power' property: \[ \left(a^{m}\right)^{n} = a^{m \cdot n} \]
  • This becomes: \[ \left(3^2\right)^2 = 3^{2\cdot 2} \]
  • Simplify further to get: \[ 3^4 = 81 \]
    Utilizing these steps and understanding the rules can make exponential equations and expressions much more manageable and less intimidating.

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    Most popular questions from this chapter

    Simplify each of the following problems by adding or subtracting. $$ 2 x^{0}+5 x^{0} $$

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