Problem 533 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[4]{5 x}\) (b) \(\sqrt[8]{9 y}\) (c) \(7 \sqrt[5]{3 z}\)

Step-by-Step Solution

Answer

(a) \( (5 x)^{1/4} \). (b) \( (9 y)^{1/8} \). (c) \( 7 (3 z)^{1/5} \).

Step 1: Understanding Rational Exponents

A rational exponent can be written in the form of a fraction. The general form is \(\sqrt[n]{a} = a^{1/n}.\)

Step 2: Converting \( \sqrt[4]{5 x} \)

Convert \sqrt[4]{5 x} to a rational exponent form. Use the formula from Step 1: \(\sqrt[4]{5 x} = (5 x)^{1/4}\).

Step 3: Converting \( \sqrt[8]{9 y} \)

Convert \( \sqrt[8]{9 y} \) to a rational exponent form. Using the same formula, \( \sqrt[8]{9 y} = (9 y)^{1/8} \).

Step 4: Converting \( 7 \sqrt[5]{3 z} \)

Convert \( 7 \sqrt[5]{3 z} \) to a rational exponent form. Since \( 7 \ \times \sqrt[5]{3 z} \) can be written as \( 7 \times (3 z)^{1/5} \), the result is \( 7 (3 z)^{1/5} \).

Key Concepts

fractional exponents

Fractional exponents offer a different way of writing radicals. By definition, a fractional exponent has the form \(a^{m/n}\), where \(a\) is the base, \(m\) is the numerator, and \(n\) is the denominator. This represents the expression \(n\)-th root of \(a\) raised to the power of \(m\).

By writing radicals using fractional exponents, mathematical expressions often become easier to manipulate, especially when applying exponent rules. Understanding this concept prepares you for complex algebraic operations that mix exponents and roots.

radicals to exponents

Converting radicals to exponents can simplify expressions and make them easier to work with. For example, the expression \(\sqrt[4]{5x}\) can be more straightforwardly expressed as \((5x)^{1/4}\). This transformation follows the general rule \(\sqrt[n]{a} = a^{1/n}\).

Using this rule:

• \(\sqrt[8]{9y}\) becomes \((9y)^{1/8}\)
• \(\sqrt[5]{3z}\) becomes \((3z)^{1/5}\)

This makes it easier to apply further algebraic operations. Additionally, it simplifies the mathematical notation, reducing the complexity of radical signs and helping to keep track of the powers.

exponent rules

Exponent rules are essential for working with both integer and fractional exponents. These rules help simplify and solve expressions more efficiently. Here are some key exponent rules you should remember:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Quotient: \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\)

Applying these rules ensures that expressions involving fractional exponents and radicals are simplified accurately. With these tools, dealing with complex expressions involving rational exponents becomes more manageable and straightforward.