Calculate (2/3)^-4: Solving Negative Power of a Fraction

Q: Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal and use the positive exponent". So $ (\frac{2}{3})^{-4} $ becomes $ (\frac{3}{2})^4 $ - we flip first, then calculate!

Q: How do I calculate $ (\frac{3}{2})^4 $ step by step?

Use the rule $ (\frac{a}{b})^n = \frac{a^n}{b^n} $. So $ (\frac{3}{2})^4 = \frac{3^4}{2^4} = \frac{81}{16} $. Calculate top and bottom separately: $ 3^4 = 81 $ and $ 2^4 = 16 $.

Q: Will the answer always be bigger than 1 with negative exponents?

When the original fraction is less than 1 (like $ \frac{2}{3} $), flipping it gives a fraction greater than 1, so yes! But if you started with $ \frac{5}{3} $, flipping would give $ \frac{3}{5} $ (less than 1).

Q: What's the difference between $ (\frac{2}{3})^{-4} $ and $ -(\frac{2}{3})^4 $ ?

Huge difference! $ (\frac{2}{3})^{-4} = \frac{81}{16} $ (positive), but $ -(\frac{2}{3})^4 = -\frac{16}{81} $ (negative). The placement of the negative sign completely changes the meaning!

Negative Exponents with Fractional Bases

$(\frac{2}{3})^{-4}=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 According to the laws of exponents, fraction (A\B) to the power of (-N)

00:06 equals fraction (B\A) to the power of (N)

00:10 Let's apply this to our question

00:13 We obtain the fraction (3\2) to the power of (4)

00:17 According to the laws of exponents, any fraction to the power of (N)

00:20 is equal to the numerator to the power of (N) divided by the denominator to the power of (N)

00:23 Let's apply this to the question

00:26 We obtained (3) to the power of 4 divided by (2) to the power of 4

00:29 Let's solve 3 to the power of 4 using the laws of exponents

00:34 Let's solve 2 to the power of 4 using the laws of exponents

00:41 Insert the results

00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(\frac{2}{3})^{-4}=\text{?}$

Step-by-step solution

We use the formula:

$(\frac{a}{b})^{-n}=(\frac{b}{a})^n$

Therefore, we obtain:

$(\frac{3}{2})^4$

We use the formula:

$(\frac{b}{a})^n=\frac{b^n}{a^n}$

Therefore, we obtain:

$\frac{3^4}{2^4}=\frac{3\times3\times3\times3}{2\times2\times2\times2}=\frac{81}{16}$

Final Answer

$\frac{81}{16}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent flips the fraction and makes exponent positive
Technique: $(\frac{2}{3})^{-4} = (\frac{3}{2})^4$ then calculate normally
Check: Multiply result by original base raised to positive power: $\frac{81}{16} \times (\frac{2}{3})^4 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Making the entire result negative
Don't think the negative exponent makes the answer negative = $-\frac{81}{16}$ ! Negative exponents only flip fractions, they don't make results negative. Always remember that negative exponents create reciprocals, not negative numbers.

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal and use the positive exponent". So $(\frac{2}{3})^{-4}$ becomes $(\frac{3}{2})^4$ - we flip first, then calculate!

How do I calculate $(\frac{3}{2})^4$ step by step?

Use the rule $(\frac{a}{b})^n = \frac{a^n}{b^n}$ . So $(\frac{3}{2})^4 = \frac{3^4}{2^4} = \frac{81}{16}$ . Calculate top and bottom separately: $3^4 = 81$ and $2^4 = 16$ .

Will the answer always be bigger than 1 with negative exponents?

When the original fraction is less than 1 (like $\frac{2}{3}$ ), flipping it gives a fraction greater than 1, so yes! But if you started with $\frac{5}{3}$ , flipping would give $\frac{3}{5}$ (less than 1).

What's the difference between $(\frac{2}{3})^{-4}$ and $-(\frac{2}{3})^4$ ?

Huge difference! $(\frac{2}{3})^{-4} = \frac{81}{16}$ (positive), but $-(\frac{2}{3})^4 = -\frac{16}{81}$ (negative). The placement of the negative sign completely changes the meaning!

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Calculate (2/3)^-4: Solving Negative Power of a Fraction

Negative Exponents with Fractional Bases

❤️ 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 does a negative exponent flip the fraction?

How do I calculate $(\frac{3}{2})^4$ step by step?

Will the answer always be bigger than 1 with negative exponents?

What's the difference between $(\frac{2}{3})^{-4}$ and $-(\frac{2}{3})^4$ ?

🌟 Unlock Your Math Potential

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Calculate (2/3)^-4: Solving Negative Power of a Fraction

Negative Exponents with Fractional Bases

❤️ 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 does a negative exponent flip the fraction?

How do I calculate (32)4 (\frac{3}{2})^4 (23​)4 step by step?

Will the answer always be bigger than 1 with negative exponents?

What's the difference between (23)−4 (\frac{2}{3})^{-4} (32​)−4 and −(23)4 -(\frac{2}{3})^4 −(32​)4?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Solve 10^(-5): Converting Negative Power to Decimal Form

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Factor the Polynomial Expression: 2a^5 + 8a^6 + 4a^3

Simplify the Nested Expression: (a^4)^6 Using Power Rules

Calculate (-5)^(-3): Solving Negative Base with Negative Exponent

Solve (3×2×4×6)^(-4): Negative Exponent Calculation

Negative Exponents

Simplify the Expression: 7^5 × 7^(-6) Using Laws of Exponents

Simplify 12⁴ × 12⁻⁶: Combining Positive and Negative Exponents

Calculate 7^(-4): Evaluating Negative Integer Exponents

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Powers of a Fraction

Solve: 10^8 + 10^-4 + (1/10)^-16 Expression with Mixed Powers

Solve the Mixed Fraction and Fourth Root Equation: 2^3/3^2 * 3^-2 * Fourth Root of 81 = ?

Solve (4²/7⁴)²: Calculating Powers of Complex Fractions

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How do I calculate $(\frac{3}{2})^4$ step by step?

What's the difference between $(\frac{2}{3})^{-4}$ and $-(\frac{2}{3})^4$ ?