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Chapter 3: Problem 4

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ 81^{3 / 4} $$

Short Answer

Expert verified
The short answer is \(81^{3/4} = 27\).

Step by step solution

01

Rewrite the power with a fractional exponent

First, we can rewrite the expression as a radical using the fractional exponent properties: \[81^{\frac{3}{4}} = \sqrt[4]{81^3}\]
02

Calculate the base raised to the numerator of the exponent

Now we need to compute \(81^3\), the base raised to the power of the numerator of the exponent: \[81^3 = 81 \times 81 \times 81 = 531441\]
03

Take the 4th root of the result

Next, we calculate the 4th root of the expression computed in the previous step (which is the denominator of the fractional exponent):\[\sqrt[4]{531441} = 27\] So, the final step by step solution is:\[81^\frac{3}{4} = \sqrt[4]{81^3} = \sqrt[4]{531441} = 27\] Therefore, \(81^{3/4} = 27\).

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