Problem 64 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the expression. $$ 2^{1 / 2} \cdot 2^{3 / 5} $$

Step-by-Step Solution

Answer

The simplified expression is \(2 \cdot 2^{1/10}\)

Step 1: Identify the problem

The problem presents an expression composed of two terms, both of which are powers of 2. The multiplication of the terms is the operation we need to simplify.

Step 2: Apply the properties of exponentiation

With the multiplication of powers rule, when multiplying two exponents with the same base, the exponents are added together. Hence, the problem can be rewritten as \(2^{1/2+3/5}\).

Step 3: Simplify the exponent

The next step is to simplify the exponent by adding together the fractions. The common denominator of 2 and 5 is 10, thus we could rewrite the fraction \(1/2\) as \(5/10\) and the fraction \(3/5\) as \(6/10\). Then add the converted fractions, the exponent becomes \(2^{5/10+6/10}\) and further simplification results in \(2^{11/10}\).

Step 4: Express exponent as mixed number

The next step is to express the fraction \(11/10\) as a mixed number. So, we get \(2^{1 1/10}\).

Step 5: Express the result as a power of a power

Using the rules of powers of a power, we could rewrite the expression as \(2^{1} \cdot 2^{1/10}\). The power of 1 to any number is that number so the expression simplifies to \(2 \cdot 2^{1/10}\).

Key Concepts

Properties of Exponents

When dealing with exponentiation, understanding the properties of exponents is crucial. Exponents serve to simplify the multiplication of similar bases and provide a concise way to handle large numbers.

• One key property is that when multiplying two numbers with the same base, you simply add their exponents. This is known as the "Product of Powers" property. For instance, in the problem provided, multiplying \(2^{1/2}\) by \(2^{3/5}\) allows us to add \(\frac{1}{2}\) and \(\frac{3}{5}\), since both terms have the base 2.
• It's important to remember that this rule only applies when the bases match. If they don't, you cannot directly apply this property without additional manipulation.

This property significantly simplifies complex expressions and is a foundational element in algebra and higher math.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, making them easier to work with or further analyze. When the task is to simplify an expression with exponentiation, applying appropriate rules or properties becomes essential.For the given exercise, once the correct exponentiation rule is identified, the expression \(2^{1/2} \cdot 2^{3/5}\) can be transformed into \(2^{1/2 + 3/5}\). Now, instead of dealing with two separate exponent terms, there is a single more manageable exponent - a sum of fractions, \(\frac{1}{2} + \frac{3}{5}\).Breaking down problems in this manner, focusing on transforming complex parts into simple fractions or whole numbers, greatly aids in managing challenging math operations. This technique is widely applicable, be it solving equations or performing calculations, and becomes second nature with practice.

Fraction Addition

Adding fractions is a fundamental skill in algebra, and becomes particularly useful when working with exponents. Unlike whole numbers, when adding fractions, their denominators must first be made the same.Consider the exercise's requirement to add \(\frac{1}{2}\) and \(\frac{3}{5}\). To add these, a common denominator is needed. Both 2 and 5 multiply into 10, so 10 becomes the common denominator:

• Convert \(\frac{1}{2}\) into \(\frac{5}{10}\) by multiplying both the numerator and denominator by 5.
• Similarly, convert \(\frac{3}{5}\) into \(\frac{6}{10}\) by multiplying both the numerator and denominator by 2.
• Add \(\frac{5}{10}\) and \(\frac{6}{10}\) to get \(\frac{11}{10}\).

Understanding and practicing how to find and apply a common denominator is crucial, not only in arithmetic but also in more advanced topics in mathematics. It serves as a foundation for accurate and efficient calculation.