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

Question

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

00:00 Solve

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 get 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 equals numerator to the power of (N) divided by denominator to the power of (N)

00:23 Let's apply to the question

00:26 We got (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 laws of exponents

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

00:41 Let's substitute the solutions

00:46 And this is the solution to the question

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:

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

$\frac{81}{16}$