Problem 45 - Algebra 2

Question

Simplify each expression. $$ \sqrt{17} \cdot \sqrt[3]{17^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 17^{7/6} \).

Step 1: Understanding the Expression

The expression given is \( \sqrt{17} \cdot \sqrt[3]{17^2} \). This is a product of two radical expressions: a square root and a cube root.

Step 2: Convert Radicals to Exponents

Convert each radical into an exponent. The square root \( \sqrt{17} \) can be written as \( 17^{1/2} \), and the cube root \( \sqrt[3]{17^2} \) can be written as \( 17^{2/3} \). The expression becomes: \( 17^{1/2} \cdot 17^{2/3} \).

Step 3: Apply the Same Base Rule for Exponents

When multiplying like bases, add the exponents. So, we add \( \frac{1}{2} \) and \( \frac{2}{3} \).

Step 4: Find Common Denominator and Add Exponents

The common denominator for the fractions \( \frac{1}{2} \) and \( \frac{2}{3} \) is 6. Convert the fractions: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{2}{3} = \frac{4}{6} \). Add them to get \( \frac{3}{6} + \frac{4}{6} = \frac{7}{6} \).

Step 5: Simplify Expression with Combined Exponent

The expression \( 17^{\frac{7}{6}} \) is the simplified form. This fraction represents the power to which 17 is raised in the combined expression.

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of the same number. For example, the expression \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \). Here, 3 is the base, and 4 is the exponent indicating how many times the base is used in the multiplication.
Exponents follow certain rules which make simplifying expressions more straightforward. Some of these rules are:

• When multiplying like bases, you add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• When dividing like bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising an exponent to another power multiplies the exponents: \( (a^m)^n = a^{m\cdot n} \).

Understanding these rules is essential in handling more complex expressions involving exponents, such as those that are related to radicals.

Radicals

Radicals are essentially the inverse operation of exponents. They express the root of a number, like square roots or cube roots. A radical symbol \( \sqrt{} \) indicates a square root, while a cube root is written as \( \sqrt[3]{} \).
It’s helpful to remember that the radical notation can be converted to exponential form. For instance:

• The square root \( \sqrt{a} \) is equivalent to \( a^{1/2} \).
• The cube root \( \sqrt[3]{a} \) is equivalent to \( a^{1/3} \).

This conversion allows radicals to be simplified using the rules of exponents. Simplification is made easier when expressions have similar roots or can be expressed with the same base, as seen in the original exercise. Such conversions are particularly useful when handling compound expressions that involve multiples of radical terms.

Fractional Exponents

Fractional exponents provide a neat way to represent roots and powers within one notation. They bridge the items found under both exponents and radicals. For example, the expression \( a^{1/2} \) represents the square root of \( a \), the same as \( \sqrt{a} \). Similarly, \( a^{2/3} \) represents the cube root of \( a^2 \).
Fractional exponents follow the same rules as regular exponents, making complex radical expressions more manageable. In the original exercise, we saw how \( \sqrt{17} \) and \( \sqrt[3]{17^2} \) are written as \( 17^{1/2} \) and \( 17^{2/3} \) respectively. This rewrite allows us to use the property of exponents governing multiplication to combine these expressions:

• Add the exponents: \( 17^{1/2} \cdot 17^{2/3} = 17^{1/2 + 2/3} = 17^{7/6} \).

Understanding fractional exponents helps make radical expressions easier to simplify by converting them into a familiar form that can be combined using basic exponent laws.