Problem 34 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 9^{-2}+9^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{244}{81}\) as a rational number in simplest form.

Step 1: Simplify the first term

The expression has two terms: \(9^{-2}\) and \(9^{\frac{1}{2}}\). Begin by simplifying \(9^{-2}\). Recall that \(a^{-b} = \frac{1}{a^b}\), so \(9^{-2} = \frac{1}{9^2}\). We know \(9^2 = 81\), thus \(9^{-2} = \frac{1}{81}\).

Step 2: Simplify the second term

Now simplify the second term \(9^{\frac{1}{2}}\). This is equivalent to the square root of 9, as \(a^\frac{1}{b} = \sqrt[b]{a}\). Therefore, \(9^{\frac{1}{2}} = \sqrt{9} = 3\).

Step 3: Add the simplified terms

Combine the simplified expressions \(\frac{1}{81}\) and \(3\). To add them, they need a common denominator. Express 3 as a fraction with denominator 81: \(3 = \frac{243}{81}\). Now, the expression is \(\frac{1}{81} + \frac{243}{81}\).

Step 4: Perform the addition

Add the fractions: \(\frac{1}{81} + \frac{243}{81} = \frac{1+243}{81} = \frac{244}{81}\).

Step 5: Simplify the result

Check if the fraction \(\frac{244}{81}\) can be simplified further. Since 244 and 81 have no common factors other than 1, the fraction is already in simplest form.

Key Concepts

Understanding Exponent Rules

Exponent rules are fundamental when dealing with any form of mathematical expressions involving powers. An exponent tells you how many times to multiply a number, which is the base, by itself. Simplifying expressions like those found in the original problem requires understanding a few key rules for exponents:

• Negative Exponents: If you encounter a negative exponent, remember that it indicates a reciprocal. For example, the expression \(9^{-2}\) means you take the reciprocal of \(9^2\). Thus, \(a^{-b} = \frac{1}{a^b}\).
• Fractional Exponents: A fraction in the exponent, like \(9^{\frac{1}{2}}\), represents a root. The denominator in the fraction (2 in this case) indicates the degree of the root, and the numerator is the power. Hence, \(a^{\frac{1}{b}}= \sqrt[b]{a}\).

In our task, we simplified \(9^{-2}\) to \(\frac{1}{81}\) by converting the negative exponent to a reciprocal, and \(9^{\frac{1}{2}}\) to \(3\) using the square root.

Mastering Fraction Addition

Adding fractions involves finding a common denominator. When fractions have different denominators, you must rewrite them so they can be added. Here's how to tackle this effectively:

• Find a Common Denominator: This is a crucial step in adding fractions. It's often beneficial to use the least common denominator to minimize further simplification. In our exercise, we needed to rewrite both \(\frac{1}{81}\) and \(3\) with the denominator 81.
• Rewriting Whole Numbers: Whole numbers like \(3\) are rewritten as fractions by giving them a denominator. In our solution, \(3\) was rewritten as \(\frac{243}{81}\).
• Add the Numerators: Once the fractions have a common denominator, simply add the numerators together while keeping the denominator the same. This results in our simplified sum, \(\frac{1 + 243}{81} = \frac{244}{81}\).

These steps ensure accurate fraction addition and make working with mixed denominators straightforward.

Simplifying Fractions

Fractions should be expressed in their simplest form, which requires that the numerator and denominator have no common factors other than 1. Simplifying a fraction makes it easier to interpret.

• Identify Common Factors: Check the numerator and the denominator for factors they might share. The greatest common factor (GCF) is used to reduce the fraction, if possible.
• Check Simplification Possibilities: In the final step of our problem, the fraction \(\frac{244}{81}\) was checked for common factors. Since 244 and 81 do not share any factors other than 1, it is already in simplest form.
• Utilize Prime Factorization: For more complicated cases, breaking numbers into their prime factors helps verify simplification. This isn't necessary for every fraction, but it's a handy tool when unsure.

Remembering these simplification steps ensures clarity in results and guarantees that values are presented as simply as possible.