Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 3^{\frac{1}{2}} \times 3^{\frac{3}{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Apply the Product of Powers Rule

To simplify the expression \(3^{\frac{1}{2}} \times 3^{\frac{3}{2}}\), we use the product of powers rule: \(a^m \times a^n = a^{m+n}\). Here \(a = 3\), \(m = \frac{1}{2}\), and \(n = \frac{3}{2}\). This gives us:\[3^{\frac{1}{2} + \frac{3}{2}}.\]

Step 2: Simplify the Exponent

Add the exponents \(\frac{1}{2}\) and \(\frac{3}{2}\) together. \(\frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2\). Therefore, the expression simplifies to:\[3^2.\]

Step 3: Evaluate the Power

Calculate \(3^2\). This is equal to \(3 \times 3\), which is \(9\). Thus, the expression \(3^{\frac{1}{2}} \times 3^{\frac{3}{2}}\) simplifies to 9.

Key Concepts

Understanding the Product of Powers Rule

The Product of Powers Rule is a fundamental exponent rule used to simplify expressions where two or more exponents with the same base are multiplied together. The rule states: \( a^m \times a^n = a^{m+n} \). This means you can add the exponents together when the bases are the same, making the calculation much easier. For example, in the expression \(3^{\frac{1}{2}} \times 3^{\frac{3}{2}}\), the base is 3 for both terms. Therefore, according to the Product of Powers Rule, you can add \( \frac{1}{2} \) and \( \frac{3}{2} \) to get the resulting exponent. This results in \( 3^{\frac{1}{2} + \frac{3}{2}} = 3^2 \).
Using this rule simplifies complex calculations by reducing multiple terms with similar bases into a single term with a new exponent. It's handy in solving algebraic problems efficiently!
By mastering this rule, you'll quickly simplify many exponent-related problems!

Simplifying Exponents

Simplifying exponents involves combining and reducing expressions with exponents into an easier form. After applying the Product of Powers Rule, the next step is to deal with the exponents themselves. In our example, we combined \( \frac{1}{2} \) and \( \frac{3}{2} \). By adding these two fractions, we got \( \frac{4}{2} \), which simplifies to 2.
This simplification process often involves adding or subtracting fractions, as we did here, or sometimes converting between improper fractions and whole numbers.

• When adding fractions, always look for a common denominator, which makes the process easier.
• Once simplified, check if further reduction (to simplest form or whole numbers) is possible.

Simplifying helps in getting clearer solutions, reducing chances of errors while solving problems. It's a crucial step in tackling mathematical expressions involving exponents!

Evaluating Powers

Once you've simplified the expression's exponent, it's time to evaluate the power. Evaluating powers means calculating the final numerical value of an expression. In the example \(3^2\), you evaluate by making the calculation \(3 \times 3\).
Here are quick tips to evaluate powers effectively:

• Know your basic powers: Memorize powers of numbers like 2, 3, and 10.
• Practice multiplying numbers to get a hang of it.
• Use patterns in powers for faster calculation, like knowing \(3^2\) equals 9 as it's \(3 \times 3\).

Once evaluated, you can confidently present the answer as the simplest form of the original expression. It's the final step in turning complex exponent expressions into clear, simple solutions!