/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 82 Simplify: \(\left(\frac{1}{3}\ri... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 82

Simplify: \(\left(\frac{1}{3}\right)^{-2}\).

Short Answer

Expert verified
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Step by step solution

01

Understand Negative Exponents

A negative exponent means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. In this case, \(\frac{1}{3}^{-2}\) becomes \(\frac{3}{1}^{2}\).
02

Simplify the Fraction

When you flip \(\frac{1}{3}\), it becomes \(\frac{3}{1}\), which is just 3.
03

Apply the Exponent

Now, apply the exponent to the simplified base: \(3^{2}\).
04

Calculate the Result

Finally, calculate \(3^{2} = 9\).

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Key Concepts

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

exponent rules
When working with exponents, there are a few key rules to keep in mind: the product of powers rule, the power of a power rule, and the negative exponent rule. A negative exponent means you need to flip the base to the reciprocal. For example, in \(\frac{1}{3}^{-2}\), the negative exponent tells us to flip the base to get \(\frac{3}{1}^{2}\). Remember: \(a^{-m} = \frac{1}{a^{m}}\). This is crucial for understanding how to simplify expressions with negative exponents.
fraction simplification
Simplifying fractions can sometimes involve flipping the fraction, especially when negative exponents are involved. For instance, when you have \(\frac{1}{3}^{-2}\), flipping it simplifies to \(\frac{3}{1}\), which is just 3. Always check if there鈥檚 a simpler form of the fraction by finding the Greatest Common Divisor (GCD) for the numerator and the denominator. In our context, the fraction flips to a whole number, making further simplification straightforward.
multiplication of integers
After simplifying the fraction and applying the exponent, you often end up with a simple multiplication problem. For \(\frac{1}{3}^{-2}\), the fraction simplifies to 3, and then you apply the exponent of 2 to get \[3^{2} = 9.\] Multiplying integers is straightforward: \(3 \times 3 = 9\). Remember to handle multiplication step-by-step, ensuring accuracy in your final answer.

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