Problem 539 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(216^{\frac{1}{3}}\) (b) \(32^{\frac{1}{5}}\) ( c\() 81^{\frac{1}{4}}\)

Step-by-Step Solution

Answer

(a) 6, (b) 2, (c) 3

Step 1: Simplify (a) \(216^{\frac{1}{3}}\)

Recognize that the exponent \(\frac{1}{3}\) represents the cube root of 216. Therefore, write this as \(\sqrt[3]{216}\). The cube root of 216 is 6, since \(6^3 = 216\). Thus, \(216^{\frac{1}{3}} = 6\).

Step 2: Simplify (b) \(32^{\frac{1}{5}}\)

Recognize that the exponent \(\frac{1}{5}\) represents the fifth root of 32. Therefore, write this as \(\sqrt[5]{32}\). The fifth root of 32 is 2, since \(2^5 = 32\). Thus, \(32^{\frac{1}{5}} = 2\).

Step 3: Simplify (c) \(81^{\frac{1}{4}}\)

Recognize that the exponent \(\frac{1}{4}\) represents the fourth root of 81. Therefore, write this as \(\sqrt[4]{81}\). The fourth root of 81 is 3, since \(3^4 = 81\). Thus, \(81^{\frac{1}{4}} = 3\).

Key Concepts

Cube Roots

Cube roots are used to find a number that, when multiplied by itself three times, gives the original number. For example, to find the cube root of 216, we ask: what number, when raised to the power of 3, equals 216? The answer is 6, because \(6^3 = 216\).Hence, \216^{\frac{1}{3}} = 6\.
Knowing how to find cube roots helps simplify expressions with exponents. It's essential in algebra and geometry.

Fifth Roots

Fifth roots are similar to cube roots but involve finding a number that, when raised to the power of 5, equals the original number. For instance, to find the fifth root of 32, you need to determine what number raised to the power of 5 gives 32. The answer is 2, as \(2^5 = 32\). Therefore, \32^{\frac{1}{5}} = 2\.
This concept is useful in advanced mathematics, including polynomial equations and number theory.

Fourth Roots

Fourth roots involve determining the number that must be raised to the power of 4 to yield a given number. For instance, to find the fourth root of 81, you need to find the number that when raised to 4 equals 81. Here, the answer is 3, because \(3^4 = 81\). Therefore, \81^{\frac{1}{4}} = 3\.
Understanding fourth roots is important for solving certain types of equations and simplifying radical expressions.

Exponentiation

Exponentiation is the process of raising a number (the base) to the power of another number (the exponent). For example, in \(6^3\), 6 is the base and 3 is the exponent, and it means multiplying 6 by itself three times: \(6 \times 6 \times 6 = 216\).
Exponentiation is fundamental in math and sciences, enabling the expression of very large or very small numbers efficiently. It's also crucial in understanding roots, as roots are the inverse operations of exponents, like \(32^{\frac{1}{5}} = 2\), where 2 is the base, and both 5 (the exponent) and \frac{1}{5}\ (the root) play complementary roles.