Question

Evaluate the expressions. (a) \(25^{3 / 2}\) (b) \(81^{1 / 2}\) (c) \(3^{-2}\) (d) \(27^{-1 / 3}\)

Short answer

(a) \(25^{3 / 2}=125\)\n(b) \(81^{1 / 2}=9\)\n(c) \(3^{-2}=0.1111\)\n(d) \(27^{-1 / 3}=0.3333\)

Step-by-step solution

Evaluate expression (a)

For the expression \(25^{3 / 2}\), it can be understood as \((25^{1 / 2})^3\), which becomes \(5^3=125\). This is based on the law of exponents \(a^{m / n} = (a^{1 / n})^m .\)

Evaluate expression (b)

For the expression \(81^{1 / 2}\), it represents square root of 81 which equals 9. This is because, according to the law of exponents, \(a^{1 / n}\) gives the nth root of a.

Evaluate expression (c)

For the expression \(3^{-2}\), it can be interpreted as \(1 / 3^{2}\), which equals \(1 / 9 = 0.1111\). This follows the law of exponents which states that any number with a negative exponent is the reciprocal of the number with a positive exponent.

Evaluate expression (d)

For the expression \(27^{-1 / 3}\), it can be represented as \(1 / (27^{1 / 3})\), which equals \(1 / 3 = 0.3333\). This is a combination of the rules for negative exponents and fractional exponents.