Evaluate (2/5)^(-2): Negative Exponent Fraction Problem

Q: Why doesn't the negative exponent make the answer negative?

The negative exponent is an instruction to take the reciprocal, not to make the result negative! Think of it as "flip the fraction" rather than "make it negative."

Q: What's the difference between $ \left(\frac{2}{5}\right)^{-2} $ and $ -\left(\frac{2}{5}\right)^{2} $ ?

The first one equals $ \left(\frac{5}{2}\right)^{2} = \frac{25}{4} $ (positive), while the second equals $ -\frac{4}{25} $ (negative). The negative sign's position makes all the difference!

Q: Can I work with the numerator and denominator separately?

No! Don't write $ \frac{5^{-2}}{2^{-2}} $ because that creates more negative exponents to deal with. Always flip the entire fraction first, then apply the positive exponent.

Q: What if the exponent was -1 instead of -2?

Same rule! $ \left(\frac{2}{5}\right)^{-1} = \left(\frac{5}{2}\right)^{1} = \frac{5}{2} $. Any negative exponent means reciprocal, regardless of the number.

Negative Exponents with Reciprocal Rules

Insert the corresponding expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the exponent laws, a fraction raised to a power(-N)

00:08 equals the reciprocal fraction raised to the opposite power (N)

00:12 We will apply this formula to our exercise

00:15 We will invert the fraction

00:19 and proceed to raise it to the opposite power (times(-1))

00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve the problem of converting $\left(\frac{2}{5}\right)^{-2}$ to positive exponents, we use the rule for negative exponents:

Negative exponent rule states:

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ — This indicates that we invert the fraction and change the sign of the exponent to positive.

Given expression: $\left(\frac{2}{5}\right)^{-2}$ .

Application: By using the rule, the negative exponent instructs us to reciprocate the fraction:

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

The positive exponent $(2)$ indicates the expression is squared. Thus, our action is complete with no further action required.

Thus, the correctly transformed expression of $\left(\frac{2}{5}\right)^{-2}$ is indeed:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means flip the fraction and make exponent positive
Technique: $\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2}$ by reciprocating
Check: Verify $\left(\frac{5}{2}\right)^{2} = \frac{25}{4}$ equals original calculation ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to the front of expression
Don't write $-\left(\frac{2}{5}\right)^2$ or $-\left(\frac{5}{2}\right)^2$ = wrong negative result! The negative exponent doesn't make the answer negative, it just means reciprocal. Always flip the fraction and keep the result positive.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

The negative exponent is an instruction to take the reciprocal, not to make the result negative! Think of it as "flip the fraction" rather than "make it negative."

What's the difference between $\left(\frac{2}{5}\right)^{-2}$ and $-\left(\frac{2}{5}\right)^{2}$ ?

The first one equals $\left(\frac{5}{2}\right)^{2} = \frac{25}{4}$ (positive), while the second equals $-\frac{4}{25}$ (negative). The negative sign's position makes all the difference!

How do I remember to flip the fraction?

Think of negative exponents as saying "I'm upside down!" When you see $\left(\frac{a}{b}\right)^{-n}$ , the fraction wants to flip to become $\left(\frac{b}{a}\right)^{n}$ .

Can I work with the numerator and denominator separately?

No! Don't write $\frac{5^{-2}}{2^{-2}}$ because that creates more negative exponents to deal with. Always flip the entire fraction first, then apply the positive exponent.

What if the exponent was -1 instead of -2?

Same rule! $\left(\frac{2}{5}\right)^{-1} = \left(\frac{5}{2}\right)^{1} = \frac{5}{2}$ . Any negative exponent means reciprocal, regardless of the number.

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Evaluate (2/5)^(-2): Negative Exponent Fraction Problem

Negative Exponents with Reciprocal Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why doesn't the negative exponent make the answer negative?

What's the difference between $\left(\frac{2}{5}\right)^{-2}$ and $-\left(\frac{2}{5}\right)^{2}$ ?

How do I remember to flip the fraction?

Can I work with the numerator and denominator separately?

What if the exponent was -1 instead of -2?

🌟 Unlock Your Math Potential

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Evaluate (2/5)^(-2): Negative Exponent Fraction Problem

Negative Exponents with Reciprocal Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why doesn't the negative exponent make the answer negative?

What's the difference between (25)−2 \left(\frac{2}{5}\right)^{-2} (52​)−2 and −(25)2 -\left(\frac{2}{5}\right)^{2} −(52​)2?

How do I remember to flip the fraction?

Can I work with the numerator and denominator separately?

What if the exponent was -1 instead of -2?

🌟 Unlock Your Math Potential

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What's the difference between $\left(\frac{2}{5}\right)^{-2}$ and $-\left(\frac{2}{5}\right)^{2}$ ?