Problem 64 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 6^{\frac{5}{2}} $$

Step-by-Step Solution

Answer

\( 6^{\frac{5}{2}} = \sqrt{7776} \)

Step 1: Identify the Exponent

The expression is given as \( 6^{\frac{5}{2}} \). Here, the exponent \( \frac{5}{2} \) is a fraction. The expression can be interpreted as a root and a power. In this case, the denominator 2 represents a square root, and the numerator 5 indicates the power.

Step 2: Convert to Radical Form

To convert \( 6^{\frac{5}{2}} \) to a radical expression, you apply the denominator of the fraction as the degree of the root and the numerator as the power. This gives us \( (\sqrt{6})^5 \).

Step 3: Simplify the Radical Form

Expressing it more clearly, \( 6^{\frac{5}{2}} \) is equivalent to \( \sqrt{6^5} \). To simplify, calculate \( 6^5 = 7776 \). Thus, the expression \( \sqrt{7776} \) is the simplest radical form of \( 6^{\frac{5}{2}} \).

Key Concepts

Fractional Exponents

Fractional exponents can seem tricky at first, but they are a helpful way to express powers and roots together in one package. When you see an expression like \( b^{\frac{m}{n}} \), it's a signal that we are combining both powering up and taking a root.

• The numerator \( m \) indicates the power you raise the number \( b \) to.
• The denominator \( n \) tells you the type of root you need to take. For example, \( n=2 \) is a square root, \( n=3 \) is a cube root, and so on.

When you deal with fractional exponents, think of them as expressing both power and roots simultaneously. For example, \( 6^{\frac{5}{2}} \) can be seen as:

• "Raise 6 to the power of 5"
• "Then take the square root of that result"

Simplifying Radicals

Simplifying radicals involves rewriting your expression into its simplest possible form while maintaining equality. When simplifying the radical \( \sqrt{6^5} \), we must carefully determine what the expression equals.

• First, compute the power: \( 6^5 = 7776 \).
• Then, simplify inside the square root: \( \sqrt{7776} \).

To simplify further might not always be easy or necessary unless it provides an integer or simpler radical. Thus, \( \sqrt{7776} \) represents the simplest radical form of \( 6^{\frac{5}{2}} \) in this context.
Remember that when working with radicals, breaking them down step-by-step ensures accuracy and a clearer understanding of the result.

Exponents and Powers

Exponents and powers form the backbone of many algebraic operations. They tell you how many times you multiply a number by itself. Understanding how exponents work is crucial in tackling problems involving powers.

• An exponent applies to a base number. The base number is multiplied by itself as many times as the exponent indicates.
• For example, \( 6^5 \) means \( 6 \times 6 \times 6 \times 6 \times 6 \).

Using rules of exponents can simplify your calculations:

• Multiplying powers with the same base adds their exponents.
• Dividing powers subtracts their exponents.
• An exponent applied to an exponent multiplies them.

Understanding these rules and how they interact makes it easier to handle more complex expressions.