Problem 552 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[5]{u^{2}}\) (b) \(\sqrt[5]{v^{8}}\) (c \(\sqrt[9]{w^{4}}\)

Step-by-Step Solution

Answer

(a) u^{2/5} (b) v^{8/5} (c) w^{4/9}

Step 1: Understanding Rational Exponents

A rational exponent is another way of expressing roots. The general form is \(\sqrt[n]{a^m}=a^{m/n}\). This means that the nth root of a number is equal to that number raised to the power of 1/n.

Step 2: Convert \(\sqrt[5]{u^{2}}\) to Rational Exponent

Using the rule from Step 1, \(\sqrt[5]{u^{2}}\) can be written as \(u^{2/5}\).

Step 3: Convert \(\sqrt[5]{v^{8}}\) to Rational Exponent

Similarly, \(\sqrt[5]{v^{8}}\) is converted to the form \(v^{8/5}\).

Step 4: Convert \(\sqrt[9]{w^{4}}\) to Rational Exponent

Finally, \(\sqrt[9]{w^{4}}\) can be written as \(w^{4/9}\).

Key Concepts

Roots

Roots are fundamental concepts in mathematics that help us understand how numbers and expressions can be simplified. When we talk about roots, we often mean taking the root of a number, such as the square root or cube root. These are special types of radicals.
For instance, the square root of a number is written as \(\backslash sqrt {a}\), and it answers the question: What number multiplied by itself gives \(\text{a}\)?

The nth root of a number a is written \(\backslash sqrt[n]{a}\). It tells you what number, multiplied by itself n times, equals \(\text{a}\). In general, roots are just another way of expressing exponents in mathematics. If you understand roots, you'll be on your way to mastering many other mathematical concepts!

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. It is written in the form \(\text{a}^{\text{b}}\), where \(\text{a}\) is the base and \(\text{b}\) is the exponent.

Exponents tell us how many times the base is multiplied by itself. For example, \(\text{2}^3\) means 2 * 2 * 2, which equals 8.

This concept extends to fractional exponents as well. When we say \(\text{a}^{\frac{m}{n}}\), it means taking the nth root of the base raised to the power of m. Exponentiation helps simplify and solve many kinds of mathematical problems more efficiently.

Fractional Exponents

Fractional exponents allow us to write roots in a compact form. Instead of using radical symbols (e.g. \(\backslash sqrt[n]{a}\)), we can use exponents. This is often more convenient and aligns with existing rules of arithmetic involving exponents.

For example, \(\backslash sqrt[5]{u^2}\) can be written as \(\text{u}^{\frac{2}{5}}\). The denominator of the exponent indicates the root, and the numerator shows the power.

Using fractional exponents makes it easier to perform operations like multiplication and division across different terms. Remember, whenever you see a radical symbol, you can always express it as a fractional exponent.