Problem 45 - 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. $$ \frac{1}{(\sqrt{5})^{3}} $$

Step-by-Step Solution

Answer

The expression \(\frac{1}{(\sqrt{5})^3}\) simplifies to \(5^{-3/2}\).

Step 1: Write the Radical as a Fractional Exponent

The expression \((\sqrt{5})^3\) is equivalent to \((5^{1/2})^3\). This is because the square root of any number is the same as raising that number to the power of \(1/2\).

Step 2: Apply the Power of a Power Rule

Using the power of a power rule \((a^m)^n = a^{m\cdot n}\), we can rewrite \((5^{1/2})^3\) as \(5^{1/2 \cdot 3} = 5^{3/2}\).

Step 3: Write the Expression with Positive Exponents

Substitute the expression back into the fraction: \(\frac{1}{(\sqrt{5})^3} = \frac{1}{5^{3/2}}\). This expression now has a positive exponent.

Step 4: Express in Simplest Form

The simplest form of the expression \(\frac{1}{5^{3/2}}\) is \((5^{-3/2})\) by using the property \(a^{-m} = \frac{1}{a^m}\). This ensures all exponents are positive.

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical expressions that involve roots. The most common type is the square root, but radical expressions can involve any root, such as cube roots or fourth roots. Understanding radical expressions is crucial as they often appear in various math problems.

• The symbol for the square root is \( \sqrt{} \). This means taking the root of a number or expression.
• If you see a cube root, it is written as \( \sqrt[3]{} \) and similarly for other roots, like the fourth root \( \sqrt[4]{} \).
• In the expression \( \sqrt{5} \), 5 is the radicand, the number under the root symbol.

Radicals can be tricky, but life gets easier once you convert them into fractional exponents. Knowing how to transition from a radical to an expression with exponents is a handy skill for simplifying complex expressions and solving equations.

Breaking Down Fractional Exponents

Fractional exponents are another way of expressing roots and powers in math. When you see an exponent that is a fraction, it represents both a power and a root.

• For example, \( x^{1/2} \) is equivalent to the square root of \( x \). The denominator signifies the root, and the numerator signifies the power.
• A more complex fractional exponent like \( x^{3/2} \) means first taking the square root \( x^{1/2} \), then raising it to the power of 3.

In the problem provided, the expression \( (\sqrt{5})^3 \) is rewritten as \( (5^{1/2})^3 \). Here, \( 5^{1/2} \) represents the square root of 5. Raising this to the power of 3 gives us \( 5^{3/2} \), which cleverly combines both root and power into a single exponent.

Simplifying Expressions with Positive Exponents

Simplifying expressions often involves rewriting them in a way that's easier to understand or work with, which includes changing negative exponents to positive ones.

• An expression like \( \frac{1}{5^{3/2}} \) can be simplified using the property \( a^{-m} = \frac{1}{a^m} \) to \( 5^{-3/2} \).
• Changing negative exponents involves the concept that a negative exponent indicates the reciprocal of that base raised to the positive of that exponent.

Ultimately, the goal is to have an expression with positive exponents because they are neater and often more straightforward to use in further calculations. This understanding makes it easier to manipulate and solve equations that model real-world scenarios.