Problem 535 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[3]{25 a}\) (b) \(\sqrt{3 b}\) (c) \(\sqrt[10]{40 c}\)

Step-by-Step Solution

Answer

(a) \( (25a)^{1/3} \) (b) \( (3b)^{1/2} \) (c) \( (40c)^{1/10} \)

Step 1: Understanding Rational Exponents

Rational exponents are another way to express roots. The nth root of a number can be expressed as the number raised to the power of 1/n. For example, \(\sqrt[n]{x} = x^{1/n}\).

Step for Part (a)

Convert \(\sqrt[3]{25 a}\) to rational exponent form by recognizing it as the cube root. So, \(\sqrt[3]{25 a} = (25a)^{1/3}\).

Step for Part (b)

Convert \(\sqrt{3 b}\) to rational exponent form by recognizing it as the square root. So, \(\sqrt{3 b} = (3b)^{1/2}\).

Step for Part (c)

Convert \(\sqrt[10]{40 c}\) to rational exponent form by recognizing it as the 10th root. So, \(\sqrt[10]{40 c} = (40c)^{1/10}\).

Key Concepts

Roots and Exponents

Roots and exponents are fundamental concepts in algebra. When we talk about roots, we mean finding a number that, when multiplied by itself a certain number of times, gives us the original number. These are often expressed as the nth root of a number. Exponents, on the other hand, tell us how many times to multiply a number by itself. They are represented as powers. Sometimes, roots can be expressed as exponents. This is known as rational exponents. For example, the square root of a number can be written as the number raised to the power of 1/2.

Cube Root

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). It is denoted as \( \sqrt[3]{x} \). Using rational exponents, we can write the cube root of a number as the number raised to the power of 1/3. For instance, \( \sqrt[3]{25a} = (25a)^{1/3} \).

Square Root

A square root of a number is a value that, when multiplied by itself, results in the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). It is represented as \( \sqrt{x} \). In terms of rational exponents, the square root can be written as the number raised to the power of 1/2. For example, \( \sqrt{3b} \) can be expressed as \( (3b)^{1/2} \). Using this form is particularly useful in algebraic operations as it simplifies the handling of roots.

Tenth Root

The tenth root of a number is a value that, when raised to the power of 10, gives the original number. This is less common but still very important in advanced mathematics. It is represented as \( \sqrt[10]{x} \). In rational exponents, we can write the tenth root as the number raised to the power of 1/10. For instance, \( \sqrt[10]{40c} \) can be written as \( (40c)^{1/10} \). Expressing roots using rational exponents helps in simplifying and solving complex equations.