Problem 52 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[3]{27 a^{3}} $$

Step-by-Step Solution

Answer

The expression \(\sqrt[3]{27 a^3}\) simplifies to \(3a\).

Step 1: Rewrite the Radical Using Fractional Exponents

The cube root of a number or expression can be rewritten using an exponent of \(\frac{1}{3}\). Thus, the expression \(\sqrt[3]{27 a^3}\) can be rewritten as \((27 a^3)^{\frac{1}{3}}\).

Step 2: Apply the Exponent to Each Term

Apply the exponent \(\frac{1}{3}\) to both the number and the variable terms separately. This gives:\[(27)^{\frac{1}{3}} \cdot (a^3)^{\frac{1}{3}}\]

Step 3: Simplify Each Term

Now, simplify each term:1. \((27)^{\frac{1}{3}} = 3\), because \(3^3 = 27\).2. \((a^3)^{\frac{1}{3}} = a^{1} = a\). This is because when raising a power to another power, you multiply the exponents: \(3 \cdot \frac{1}{3} = 1\).

Step 4: Combine the Simplified Terms

Combine the simplified terms to express the expression in simplest form:\[3 \cdot a = 3a\]

Key Concepts

Fractional Exponents

Fractional exponents are a unique way to express roots, such as square roots, cube roots, and beyond, in terms of powers. This concept can simplify complex expressions, make calculations more straightforward, and help in algebraic manipulations. For instance, the cube root of a number or expression, like \( \sqrt[3]{27} \), can be rewritten in terms of a fractional exponent as \( 27^{\frac{1}{3}} \).

• The numerator of the fractional exponent tells you the power to which the number is raised.
• The denominator indicates the root. So, in \( 27^{\frac{1}{3}} \), "1" is the power, and "3" is the cube root.

Understanding fractional exponents can significantly simplify radical expressions, making problems easier to solve and expressions more consistent when performing operations such as multiplication and division.

Cube Root

The cube root is a specific type of root used to reflect a number that, when raised to the power of 3, results in the original number. It's like the fruit of multiplication where the number is multiplied by itself three times. For example, the cube root of 27, written as \( \sqrt[3]{27} \), results in 3, because \(3 \times 3 \times 3 = 27\).

When you have an expression that requires the cube root, like \( \sqrt[3]{27a^3} \), it can be helpful to rewrite this using fractional exponents. By rewriting it as \( (27a^3)^{\frac{1}{3}} \), you can apply the exponent to each individual term within the expression: \((27)^{\frac{1}{3}}\) and \((a^3)^{\frac{1}{3}}\).

In this approach, taking the cube root becomes a matter of applying the fractional exponent \(\frac{1}{3}\) to both components, leading to a straightforward and effective way of simplifying mathematical expressions.

Simplifying Expressions

Simplifying expressions, especially those involving radicals and powers, helps in reducing complex problems to their simplest form. This makes calculations more manageable and solutions more evident. To simplify an expression like \((27a^3)^{\frac{1}{3}}\), it's important to break it into more digestible parts:

• Solve \((27)^{\frac{1}{3}}\): Recognizing 27 as a perfect cube of 3 (since \(3^3 = 27\)), simplifies directly to 3.
• Solve \((a^3)^{\frac{1}{3}}\): Applying the power rule \((a^m)^n = a^{m \cdot n}\), the expression reduces to \(a^{3 \cdot \frac{1}{3}} = a^1 = a\).

Finally, combine the results to yield the simplest form of the expression: \(3a\). This entire process showcases how manipulating powers and roots through simplification can transform seemingly complex combinations into those that are easier to manage and solve.