Problem 64 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ 5^{3 / 5} \cdot 5^{1 / 10} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 3.65.

Step 1: Apply the Product of Powers Rule

When multiplying two exponents with the same base, add the exponents. In this expression, both terms have the base 5. So, add the exponents: \( \frac{3}{5} + \frac{1}{10} \).

Step 2: Find a Common Denominator

To add \( \frac{3}{5} \) and \( \frac{1}{10} \), convert them to have a common denominator. The least common denominator for 5 and 10 is 10. Rewrite \( \frac{3}{5} \) as \( \frac{6}{10} \).

Step 3: Add the Exponents

Now that both fractions have the same denominator, add \( \frac{6}{10} + \frac{1}{10} = \frac{7}{10} \).

Step 4: Substitute Back the Exponent

The expression simplifies to \( 5^{7/10} \).

Step 5: Approximate the Result

Calculate \( 5^{7/10} \) using a calculator. This can be approximated to \(5^{0.7} \approx 3.65\).

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a handy tool in mathematics that simplifies calculations involving exponents. This rule states that when you multiply two powers with the same base, you simply add the exponents.

Let's break it down further:

• If you have an expression of the form \( a^m \times a^n \), you use the Product of Powers Rule to rewrite it as \( a^{m+n} \).
• This rule works because multiplying numbers with the same base is like repeatedly multiplying that base by itself.

For example, in the expression \( 5^{3/5} \cdot 5^{1/10} \), both terms have a base of 5. To simplify, we add the exponents, \( \frac{3}{5} + \frac{1}{10} \).

This leads us conveniently into the next concept, as adding fractions requires finding a common denominator.

Fraction Addition

Adding fractions is a fundamental skill in math, crucial for correctly applying the Product of Powers Rule here. To add fractions:

• First, find a common denominator, the smallest number that both denominators divide into evenly.
• Once fractions have a common denominator, you can add their numerators directly, keeping the denominator the same.

In our exercise, we need to add \( \frac{3}{5} \) and \( \frac{1}{10} \). Here's how it's done:

• The least common denominator of 5 and 10 is 10.
• Convert \( \frac{3}{5} \) to an equivalent fraction with this denominator: \( \frac{3}{5} = \frac{6}{10} \).
• Now, add \( \frac{6}{10} + \frac{1}{10} = \frac{7}{10} \).

This simplified fraction, \( \frac{7}{10} \), becomes the new exponent in our expression.

Exponentiation

Exponentiation is the process of raising a number to a power, an expression written as \( a^n \). This process is like multiplying the base by itself a number of times specified by the exponent. In cases of non-integer exponents, like the fractional exponent in our solution, it implies a more complex mathematical operation.

Here's what happens with fractional exponents:

• A fractional exponent like \( a^{m/n} \) represents the \( n \)-th root of \( a^m \), or \( (a^m)^{1/n} \).
• Such exponents allow us to find roots of numbers, as well as raise numbers to specific powers.

In our example, we simplified the expression to \( 5^{7/10} \). With a calculator, you can approximate this value as \( 5^{0.7} \) or about 3.65.

Understanding exponentiation is vital for simplifying complex expressions, especially when dealing with roots and real-world applications.