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

(23)−4=? (\frac{2}{3})^{-4}=\text{?}

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Step-by-step video solution

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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
1

Understand the problem

(23)−4=? (\frac{2}{3})^{-4}=\text{?}

2

Step-by-step solution

We use the formula:

(ab)−n=(ba)n (\frac{a}{b})^{-n}=(\frac{b}{a})^n

Therefore, we obtain:

(32)4 (\frac{3}{2})^4

We use the formula:

(ba)n=bnan (\frac{b}{a})^n=\frac{b^n}{a^n}

Therefore, we obtain:

3424=3×3×3×32×2×2×2=8116 \frac{3^4}{2^4}=\frac{3\times3\times3\times3}{2\times2\times2\times2}=\frac{81}{16}

3

Final Answer

8116 \frac{81}{16}

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\( 112^0=\text{?} \)

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