Problem 23 - Algebra 2

Question

Write each expression in exponential form. $$(\sqrt[3]{a})^{2}$$

Step-by-Step Solution

Answer

So, the expression \((\sqrt[3]{a})^{2}\) in exponential form is \(a^{2/3}\).

Step 1: Understand the concept of roots and exponents

Firstly, a cubed root can be written as a power of one-third, which means \(\sqrt[3]{a}\) can be written as \(a^{1/3}\). Next, remember the power of a power rule, which states that when you have an exponent raised to another exponent, you multiply the exponents.

Step 2: Rewrite the root in exponential form

Rewrite \(\sqrt[3]{a}\) as \(a^{1/3}\) to get \((a^{1/3})^{2}\).

Step 3: Apply the power-of-a-power rule

Next, apply the power of a power rule. Multiply the exponents to get \(a^{(1/3)*2}\).

Key Concepts

Roots and Radicals

Roots and radicals help us express the inverse operation of exponentiation. The cubed root, for instance, is a specific type of radical that is commonly represented using the symbol \(\sqrt[3]{}\). This symbol denotes what we need to multiply by itself three times to obtain the original number. So, when we have \(\sqrt[3]{a}\), it essentially means the number which, when cubed, gives \(a\).

When we convert roots into exponential terms, it aids in simplifying complex equations, thus lowering the potential for errors during calculations.

• Cubed root (\(\sqrt[3]{a}\)) can be expressed in its exponential form as \(a^{1/3}\).
• This transformation of radicals into fractional exponents helps streamline manipulations involving roots and powers.

Understanding roots and radicals is essential, as it sets the foundation for more advanced algebraic manipulations.

Exponential Form

The exponential form is a concise way to write repeated multiplication of the same factor. In mathematics, expressing a number in exponential form involves rewriting it as a base raised to an exponent. In our example, the base is \(a\), and the exponent \(\frac{1}{3}\) comes from converting the cubed root.

This form is not only compact but also makes it easier to apply mathematical rules, such as the power of a power rule. Here's why it matters:

• Reduces Complexity: Exponential form simplifies complex expressions, making them easier to work with.
• Foundation for Calculations: Using exponential form allows students to apply rules, like multiplying and dividing exponents, more systematically.

Writing radicals like \((\sqrt[3]{a})^{2}\) in exponential form, such as \((a^{1/3})^{2}\), illustrates these advantages, leading to more streamlined calculations and a deeper understanding of mathematical relationships.

Power of a Power Rule

The power of a power rule is a significant property in exponentiation, crucial for simplifying expressions where exponents are raised to further exponents. According to this rule, you multiply the exponents when an exponent is raised to another exponent.

For example, when we encounter \((a^{1/3})^{2}\), applying the power of a power rule means that you multiply \(1/3\) by \(2\). This leads to \(a^{2/3}\). Here’s how it works:

• Multiply the exponents: In our case, \((1/3) \times 2 = 2/3\).
• Result: The expression \((a^{1/3})^{2}\) becomes \(a^{2/3}\).

This rule significantly simplifies expressions and is fundamental when dealing with powers of powers, making it a crucial concept in algebra and higher-level math. Understanding this rule enhances the ability to tackle various algebraic expressions elegantly and effectively.