Problem 22 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The equivalent exponential form of \(\sqrt[3]{a^{2}}\) is \(a^{2/3}\).

Step 1: Converting Radical to Exponential Form

To convert a radical to an exponential form, the formula \(\sqrt[n]{a^m} = a^{m/n}\) is used. Here, \(\sqrt[3]{a^{2}}\) represents the cube root of \(a^2\). Substitute \(n=3\) and \(m=2\) into the given formula.

Step 2: Substitution

Plugging the values \(n=3\) and \(m=2\) into the equivalent exponential form will yield \(a^{m/n}=a^{2/3}\).

Key Concepts

Radicals

In mathematics, radicals refer to expressions that involve roots, such as square roots or cube roots. The radical symbol, \(\sqrt{}\), is used to denote a root. Radicals help simplify complex expressions by indicating how many times a number must be multiplied by itself to achieve another number. For example, \(\sqrt{9}\) means "what number times itself equals 9?" and the answer is 3.

• Understanding Radicals: The number under the radical sign is called the radicand, and the small number outside the radical symbol, known as the index, tells us which root to take.
• Examples of Radicals: \(\sqrt{16} = 4\) and \(\sqrt[3]{27} = 3\).

In the expression \(\sqrt[3]{a^2}\), "\(a^2\)" is the radicand, and 3 is the index, which means we are looking for the cube root.

Exponents

Exponents are a mathematical shorthand for expressing repeated multiplication. When you see a number like \(a^n\), it means you multiply "a" by itself "n" times. Here, "a" is the base, and "n" is the exponent.
In the expression \(a^2\), the base "a" is multiplied by itself once (since 2 - 1 = 1), giving you \(a \times a = a^2\).

• Exponential Growth: As the exponent increases, the value of the expression can grow very quickly.
• Exponent Rules: Common exponent rules include multiplying powers with the same base by adding exponents, and power of a power rule (\( (a^m)^n = a^{m \cdot n} \)).

Exponents simplify mathematical expressions and make it easier to perform calculations involving large numbers.

Rational Exponents

Rational exponents are a way to express roots using exponents, where the exponent is a fraction. The expression \(a^{m/n}\) is equivalent to \(\sqrt[n]{a^m}\). This means instead of using a radical symbol, you can use a fraction as an exponent to express the same idea.
For example, \(a^{2/3}\) indicates the cube root of \(a^2\). Here:

• The numerator (2) tells us the power of "a" inside the radical.
• The denominator (3) indicates the type of root (cube root in this case).

With rational exponents, you can convert between radical and exponential forms easily. Converting \(\sqrt[3]{a^2}\) to \(a^{2/3}\) makes it simpler to apply additional exponent rules, ultimately streamlining computations.