/*! 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 39 $$\left(\frac{2}{5}\right)^{-2}$... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 39

$$\left(\frac{2}{5}\right)^{-2}$$

Short Answer

Expert verified
\(\left( \frac{2}{5} \right)^{-2} = \frac{25}{4}\)

Step by step solution

01

Understand the Negative Exponent

A negative exponent signifies that the base should be reciprocated and the exponent applied to the reciprocal. In this case, we have \((\frac{2}{5})^{-2}\).
02

Reciprocal of the Base

Take the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\).
03

Apply the Positive Exponent

Raise the reciprocal to the positive exponent 2: \(\left( \frac{5}{2} \right)^2\).
04

Simplify the Exponentiation

Simplify \(\left( \frac{5}{2} \right)^{2}\): \(((5^2) / (2^2)) = \frac{25}{4}\).

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Key Concepts

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

Understanding Reciprocals
A reciprocal of a number is quite simple. It just means flipping the numerator and the denominator of a fraction. This concept comes in handy, especially when dealing with negative exponents. Remember, the reciprocal of any fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). For instance, the reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\). When you see a negative exponent like \((\frac{2}{5})^{-2}\), you first find the reciprocal.
Simplifying Exponents
Simplifying exponents can make complicated problems easier. When you encounter a negative exponent, flip the fraction first. For example, with \((\frac{2}{5})^{-2}\), you flip \(\frac{2}{5}\) into \(\frac{5}{2}\). Then, apply the positive exponent to the new base. So, raising \(\frac{5}{2}\) to the power of 2 means calculating \(\frac{5}{2} \times \frac{5}{2} = \frac{25}{4}\). That gives us the simplified form.
Working with Fractions in Exponents
Fractions can seem tricky at first, particularly when dealing with exponents. Here鈥檚 a breakdown using our example \((\frac{2}{5})^{-2}\):
- First, recognize the negative exponent, which tells us to use the reciprocal.
- After flipping \(\frac{2}{5}\) to \(\frac{5}{2}\), we apply the positive exponent.
- This means we multiply \(\frac{5}{2}\) by itself, i.e., \(\frac{5}{2} \times \frac{5}{2} = \frac{25}{4}\).
Don't rush through fractions 鈥� take each step slowly and double-check your work to avoid mistakes.

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