Problem 63 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ 2^{1 / 2} \cdot 2^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 2.30.

Step 1: Understand the expression

The expression we have is a multiplication of two exponential terms, \(2^{1/2}\) and \(2^{2/3}\). We need to multiply these two exponential terms together.

Step 2: Apply properties of exponents

To multiply terms with the same base, use the property of exponents: \(a^m \cdot a^n = a^{m+n}\). In this case, both terms have the base 2. So, we apply the property: \(2^{1/2} \cdot 2^{2/3} = 2^{1/2 + 2/3}\).

Step 3: Find a common denominator for the fractions

Add the exponents \(1/2\) and \(2/3\) by first finding a common denominator, which is 6. Rewriting the fractions: \(1/2 = 3/6\) and \(2/3 = 4/6\).

Step 4: Add the exponents

Add \(3/6\) and \(4/6\) to get \(7/6\). Thus, the expression becomes \(2^{7/6}\).

Step 5: Approximate the result

Convert \(2^{7/6}\) to a decimal using a calculator. This gives approximately 2.297. Rounded to the nearest hundredth, the result is 2.30.

Key Concepts

Properties of Exponents

Exponents are a way to express repeated multiplication. There are several properties of exponents that can help simplify complex expressions:

• When you multiply terms with the same base, you add the exponents. This is known as the product of powers property: \( a^m \cdot a^n = a^{m+n} \).
• When you divide terms with the same base, you subtract the exponents: \( a^m / a^n = a^{m-n} \).
• Any number raised to the power of zero is 1: \( a^0 = 1 \).
• Raising a power to another power involves multiplying the exponents: \((a^m)^n = a^{m \cdot n} \).

Understanding these properties will make solving expressions involving exponents easier and more intuitive.

Multiplication of Exponential Terms

Multiplying exponential terms becomes straightforward when all the bases are the same. For the expression \(2^{1/2} \cdot 2^{2/3}\), the base "2" remains unchanged.
Apply the property of exponents, which states that you add the exponents when multiplying like bases. This transforms the problem to \(2^{1/2 + 2/3}\).
It's crucial to note that this property only applies when the terms share the same base. Other operations require different rules.

Fraction Addition

Adding fractions involves finding a common denominator. This allows you to combine the numerators easily.

• First, identify the least common multiple (LCM) of the denominators. For \(1/2 + 2/3\), the LCM of 2 and 3 is 6.
• Rewrite each fraction so that both have the common denominator. Convert \(1/2\) to \(3/6\) and \(2/3\) to \(4/6\).
• Add the new numerators: \(3/6 + 4/6 = 7/6\).

This process is essential in the evaluation of exponential expressions or any other math involving fractions.

Approximating Results

Once you've manipulated the expression, sometimes you'll end up with an irrational number or a difficult exponential form. Approximating such results can provide a more practical answer.
Using a calculator, raise a base to a fractional exponent, like \(2^{7/6}\). This gives a decimal approximation.
In this case, \(2^{7/6} \approx 2.297\). By rounding to the nearest hundredth, the result becomes 2.30. Rounding numbers makes them easier to use in real-world applications and helps present answers clearly.