(2/3)^-4 X (27/8)^-2

2 min read Jun 16, 2024

Simplifying the Expression (2/3)^-4 x (27/8)^-2

This article will guide you through simplifying the expression (2/3)^-4 x (27/8)^-2. We will use the properties of exponents to break down the problem step-by-step.

Understanding Negative Exponents

The key to solving this problem lies in understanding how negative exponents work. A negative exponent indicates the reciprocal of the base raised to the positive version of the exponent. In other words:

a^-n = 1 / a^n

Applying the Rule to Our Expression

Let's apply this rule to our expression:

(2/3)^-4 = 1 / (2/3)^4
(27/8)^-2 = 1 / (27/8)^2

Now our expression becomes:

1 / (2/3)^4 x 1 / (27/8)^2

Simplifying the Fractions

Next, we need to simplify the fractions with positive exponents:

(2/3)^4 = (2/3) x (2/3) x (2/3) x (2/3) = 16/81
(27/8)^2 = (27/8) x (27/8) = 729/64

Our expression now looks like:

1 / (16/81) x 1 / (729/64)

Inverting and Multiplying

To divide by a fraction, we invert and multiply. This gives us:

(81/16) x (64/729)

Simplifying the Multiplication

Finally, we multiply the numerators and denominators:

(81 x 64) / (16 x 729)

This simplifies to:

(9 x 8) / (2 x 9)

Further simplification yields:

4 / 1 = 4

Conclusion

Therefore, the simplified value of the expression (2/3)^-4 x (27/8)^-2 is 4.