Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 2\left[(3)^{-2}+(4)^{-2}\right]^{-\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified rational number is \(\frac{24}{5}\).

Step 1: Evaluate the Inner Exponents

Start by calculating each of the expressions with negative exponents inside the brackets. For \((3)^{-2}\), this is the same as \(\frac{1}{3^2} = \frac{1}{9}\). Similarly, for \((4)^{-2}\), this becomes \(\frac{1}{4^2} = \frac{1}{16}\).

Step 2: Add the Fractions Inside the Brackets

Next, add the fractions \(\frac{1}{9}\) and \(\frac{1}{16}\), which requires a common denominator. The least common denominator of 9 and 16 is 144. Convert each fraction: \(\frac{1}{9} = \frac{16}{144}\) and \(\frac{1}{16} = \frac{9}{144}\). Thus, the sum is \(\frac{16}{144} + \frac{9}{144} = \frac{25}{144}\).

Step 3: Apply the Negative Exponent

The expression inside the brackets with the sum, \(\left(\frac{25}{144}\right)^{-rac{1}{2}}\), needs to be evaluated by flipping the fraction and then taking the square root. Flipping gives \(\left(\frac{144}{25}\right)\). Taking the square root results in \(\left(\sqrt{\frac{144}{25}}\right) = \frac{12}{5}\).

Step 4: Multiply by 2

Finally, multiply the result from Step 3 by 2 as per the outer part of the original expression: \(2 \times \frac{12}{5} = \frac{24}{5}\). This fraction is already in simplest form.

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but once you understand the concept, they become much easier to handle. When you see a negative exponent, like \(a^{-n}\), it means you are taking the reciprocal of the base, raised to the positive exponent.
For example:

• \(3^{-2}\) is calculated as \(\frac{1}{3^2}\), which equals \(\frac{1}{9}\).
• Similarly, \(4^{-2}\) is \(\frac{1}{4^2} = \frac{1}{16}\).

When dealing with negative exponents inside fractions, always remember: negative exponents flip the numbers, turning them into fractions. It makes the division process very intuitive as it all boils down to finding reciprocals and understanding fraction rules.

Fraction Addition

Adding fractions is a fundamental concept in mathematics and requires a common denominator. This means both fractions must be expressed with the same bottom number before they can be directly added.
Here’s how you do it:

• Identify the least common denominator (LCD). For \(\frac{1}{9}\) and \(\frac{1}{16}\), the LCD is 144 because 144 is the smallest number that is a multiple of both 9 and 16.
• Convert each fraction to an equivalent fraction with the LCD as the new denominator. This conversion uses multiplication of both the numerator and denominator to make them equivalent fractions:
  • \(\frac{1}{9} = \frac{16}{144}\)
  • \(\frac{1}{16} = \frac{9}{144}\)
• After converting, simply add the numerators while keeping the common denominator: \(\frac{16 + 9}{144} = \frac{25}{144}\).

This systematic approach helps with clarity and accuracy when adding fractions.

Square Roots

Taking the square root of a fraction is similar to finding the square root of individual numbers. When asked to take the square root of \(\frac{25}{144}\), you take the square root of the numerator and the square root of the denominator separately.
The calculation looks like this:

• Find \(\sqrt{25}\), which is 5.
• Find \(\sqrt{144}\), which is 12.

Putting these together gives \(\sqrt{\frac{25}{144}} = \frac{5}{12}\).When square roots combine with fractions, remember this rule: the square root of a fraction is the fraction of the square roots. This makes computations straightforward and helps organize your work effectively.

Simplifying Fractions

To simplify a fraction, find the greatest common factor (GCF) of the numerator and the denominator, and divide both by it. A fraction is in simplest form if no number other than 1 can divide both the numerator and the denominator.
For instance:

• In the expression \(\frac{24}{5}\), the numerator (24) and the denominator (5) do not have any common factors other than 1.
• Therefore, \(\frac{24}{5}\) is already simplified.

Simplifying fractions ensures the solution is expressed in the most concise manner. It is the step that ensures clarity and finality in problem-solving, and it often highlights the most "practical" representation of a number's value.