-9%5E(-3)times-16%5E((1)%2F(4)))%2F(6%5E(-2))times((1)%2F(27))%5E(-(4)%2F(3)).jpeg "(vi) 9^(-3)times 16^((1)/(4)))/(6^(-2))times((1)/(27))^(-(4)/(3))")
Simplifying the Expression: (9^(-3) times 16^((1)/(4)))/(6^(-2)) times ((1)/(27))^(-(4)/(3))
This expression involves several exponents and fractions, making it look complicated. Let's break it down step by step to simplify it.
Understanding the Properties of Exponents
Before we start, let's recall some important properties of exponents:
- Negative Exponent: x^(-n) = 1/x^n
- Fractional Exponent: x^(m/n) = (x^m)^(1/n) = (x^(1/n))^m
- Product of Powers: x^m * x^n = x^(m+n)
- Quotient of Powers: x^m / x^n = x^(m-n)
Applying the Properties
Now let's apply these properties to simplify the expression:
-
Dealing with Negative Exponents:
- 9^(-3) = 1/9^3
- 6^(-2) = 1/6^2
-
Simplifying Fractional Exponents:
- 16^(1/4) = (2^4)^(1/4) = 2^(4/4) = 2
- (1/27)^(-4/3) = (27^(1/3))^4 = 3^4
-
Substituting the Simplified Values:
The expression now becomes: (1/9^3 * 2) / (1/6^2) * 3^4
-
Simplifying Further:
- (1/9^3 * 2) = 2/729
- (1/6^2) = 1/36
- 3^4 = 81
The expression becomes: (2/729) / (1/36) * 81
-
Division by a Fraction:
Dividing by a fraction is equivalent to multiplying by its inverse. Therefore:
(2/729) / (1/36) = (2/729) * 36 = 72/729
-
Final Multiplication:
(72/729) * 81 = 5832/729
Final Answer
Therefore, the simplified form of the given expression is 5832/729.