Problem 527 - Elementary Algebra

Question

In the following exercises, write as a radical expression. (a) \(g^{\frac{1}{7}}\) (b) \(h^{\frac{1}{5}}\) (c) \(j^{\frac{1}{25}}\)

Step-by-Step Solution

Answer

(a) \root{7}{g}\; (b) \root{5}{h}\; (c) \root{25}{j}\.

Step 1: Understanding Radical Expressions

A radical expression can be written using the notation \(\root{n}{a}\) where \(a^{\frac{1}{n}}\) converts to \(\root{n}{a}\). This is based on the property \(a^{\frac{1}{n}} = \root{n}{a}\).

Step 2: Convert \(g^{\frac{1}{7}}\) to a Radical Expression

Following the conversion rule, \(g^{\frac{1}{7}}\) converts to the 7th root of \(g\). Hence, \(g^{\frac{1}{7}} = \root{7}{g}\).

Step 3: Convert \(h^{\frac{1}{5}}\) to a Radical Expression

Using the conversion rule, \(h^{\frac{1}{5}}\) converts to the 5th root of \(h\). Therefore, \(h^{\frac{1}{5}} = \root{5}{h}\).

Step 4: Convert \(j^{\frac{1}{25}}\) to a Radical Expression

Applying the conversion rule, \(j^{\frac{1}{25}}\) converts to the 25th root of \(j\). Thus, \(j^{\frac{1}{25}} = \root{25}{j}\).

Key Concepts

Exponent Rules

Understanding exponent rules is key to working with radical expressions. Exponents are used to indicate repeated multiplication. For example, in the expression \(a^3\), \(a\) is multiplied by itself three times: \(a \times a \times a\).

There are a few essential rules you need to know about exponents:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power Rule: \( (a^m)^n = a^{mn} \)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
• Zero Exponent Rule: \(a^0 = 1\), provided that \(a eq 0\)

Specifically, when you deal with radical expressions, you often encounter fractional exponents. For instance, \(a^{\frac{1}{n}}\) represents the \(n\)th root of \(a\). This is the basis for converting exponents to radicals.

Roots

Roots help us find a number which, when multiplied by itself a certain number of times, gives us the original number. The square root is the most common type, but you can have any root like cube roots or, as seen in our exercise, 7th, 5th, and 25th roots.

In mathematical notation, the \(n\)th root of \(a\) is expressed as \( \sqrt[n]{a} \). For example:

• The square root of \(a\) is \( \sqrt{a} \) or \(a^{\frac{1}{2}}\)
• The cube root of \(a\) is \( \sqrt[3]{a} \) or \(a^{\frac{1}{3}}\)
• The 5th root of \(a\) is \( \sqrt[5]{a} \) or \(a^{\frac{1}{5}}\)

Thus, \( g^{\frac{1}{7}} \) becomes the 7th root of \(g\), \( \sqrt[7]{g} \). Similarly, \( h^{\frac{1}{5}} \) is the 5th root of \(h\), \( \sqrt[5]{h} \), and \( j^{\frac{1}{25}} \) is \( \sqrt[25]{j} \).

Converting Expressions

Converting between different forms of expressions is very useful in algebra. When it comes to exponents and radicals, it's crucial to know how to switch between these formats for simplification and solving purposes.

To convert a radical expression into an exponent form, use the rule: \( \sqrt[n]{a} = a^{\frac{1}{n}} \). Hence:

• \( \sqrt{a} = a^{\frac{1}{2}} \)
• \( \sqrt[3]{a} = a^{\frac{1}{3}} \)
• \( \sqrt[4]{a} = a^{\frac{1}{4}} \)
• \( \sqrt[5]{a} = a^{\frac{1}{5}} \)

Conversely, converting from exponents to radicals requires reversing this process. For example, turning \( g^{\frac{1}{7}} \) into \( \sqrt[7]{g} \).

By mastering these conversions, you can handle a wide range of algebraic problems with greater ease.