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

00:06 Let's solve this problem together.

00:10 Remember, with exponents, fraction A over B to the power of negative N

00:15 is the same as fraction B over A to the power of positive N.

00:21 Let's apply this rule now.

00:24 We have fraction 3 over 2 to the power of 4.

00:29 Any fraction raised to N means the top number to N

00:34 divided by the bottom number to N.

00:38 Let's apply it.

00:40 We get 3 to the fourth power over 2 to the fourth power.

00:45 Now, let's calculate 3 to the fourth power.

00:49 Then, calculate 2 to the fourth power.

00:53 Make sure to plug in the answers.

00:57 And there you have 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

Solve the following problem:

$$ 1^3= $$

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?

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