Problem 106 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -\frac{1}{9}\left(\frac{1}{4}\right)+\left(-\frac{1}{6}\right)^{2} $$

Step-by-Step Solution

Answer

0

Step 1: Evaluate the First Term

Start by evaluating the first term \(-\frac{1}{9}\left(\frac{1}{4}\right)\). Multiply the fractions: \(-\frac{1}{9} \times \frac{1}{4} = -\frac{1}{36}\).

Step 2: Evaluate the Exponent in the Second Term

Now, evaluate the second term \(\left(-\frac{1}{6}\right)^{2}\). Remember that when you square a negative number, the result is positive. Thus, \( \left(-\frac{1}{6}\right)^{2} = \left(-\frac{1}{6}\right) \times \left(-\frac{1}{6}\right) = \frac{1}{36}\).

Step 3: Add the Two Results Together

Add the results from Step 1 and Step 2: \(-\frac{1}{36} + \frac{1}{36}\). These terms cancel each other out, resulting in 0.

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, we're discussing the process of finding the value of an expression by performing the operations within it. This involves understanding the operators involved such as addition, subtraction, multiplication, division, and even exponents.

The key is to tackle one operation at a time, following the correct order. For example, consider the expression we evaluated:

• First, you need to handle the multiplication and exponentiation before moving on to addition or subtraction.
• Finally, put together the results of each part of the expression to come up with a final value.

In more complex expressions, it's useful to break down the expression into smaller parts, evaluate each part, and then combine those results step by step. By being systematic in your approach, even complicated expressions can be simplified into something manageable.

Multiplying Fractions

Multiplying fractions might seem tricky, but it's straightforward once you know the steps. When multiplying two fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.

Here’s how you do it:

• For example, to evaluate evaluate \(-\frac{1}{9} \times \frac{1}{4}\), you multiply -1 with 1 to get -1, and 9 with 4 to get 36, resulting in \(-\frac{1}{36}\).
• Remember, if one fraction is negative, the result will also be negative.
• It's also important to simplify the result whenever possible, but in this case, \(-\frac{1}{36}\) is already in its simplest form.

Multiplying fractions is an important skill that comes in handy in various topics, so it's worth practicing until you're comfortable.

Exponents with Fractions

Dealing with exponents can add an extra step into evaluating fractions, but it’s manageable once you understand the rules. When you see a fraction raised to a power, each part of the fraction—both numerator and denominator—must be raised to that power.

Let’s break it down with an example from our exercise:

• For \(\left(-\frac{1}{6}\right)^{2}\), we square both -1 and 6. This means multiplying -1 by -1 to get 1, and 6 by 6 to get 36.
• The negative times a negative results in a positive, giving us \(\frac{1}{36}\).
• This positive result is crucial, as the negative sign in the base disappears when pushed through an even exponent (like 2).

Understanding these rules will make working with exponents applied to fractions much more intuitive. Remember: even exponents result in a positive outcome regardless of whether you start with a negative base.