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Chapter 4: Problem 36

Simplify the expression. $$\left(\frac{1}{4}\right)\left(\frac{1}{6}\right)+\left(-\frac{5}{6}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \( \frac{53}{72} \).

Step by step solution

01

Multiply Fractions

The first part of the expression is \( \left(\frac{1}{4}\right)\left(\frac{1}{6}\right) \). To multiply fractions, multiply the numerators and then the denominators.\[ \frac{1 \times 1}{4 \times 6} = \frac{1}{24} \]
02

Square the Negative Fraction

The next part is \( \left(-\frac{5}{6}\right)^{2} \). When squaring a fraction, square both the numerator and the denominator. Remember, \((-a)^2 = a^2\). \[ \left(\frac{-5}{6}\right)^2 = \frac{(-5)^2}{6^2} = \frac{25}{36} \]
03

Add the Results

Now, add the results from steps 1 and 2. To add fractions \( \frac{1}{24} + \frac{25}{36} \), we must find a common denominator. The least common multiple of 24 and 36 is 72. Convert each fraction to have a denominator of 72. \[ \frac{1}{24} = \frac{3}{72} \quad (by \ multiply \ numerator \ and \ denominator \ by \ 3)\] \[ \frac{25}{36} = \frac{50}{72} \quad (by \ multiply \ numerator \ and \ denominator \ by \ 2) \] Add the two fractions: \[ \frac{3}{72} + \frac{50}{72} = \frac{53}{72} \]
04

Verify Simplicity

Check if \( \frac{53}{72} \) can be simplified further by finding the greatest common divisor (GCD) of 53 and 72. Since 53 is a prime number and does not divide 72, the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Fractions
Multiplying fractions is a straightforward process once you understand the rule of multiplying the numerators together and the denominators together. Let鈥檚 break it down with a simple example.
  • Start by looking at the fractions you want to multiply: \( \frac{1}{4} \) and \( \frac{1}{6} \).
  • Multiply the top numbers (numerators):
    \( 1 \times 1 = 1 \).
  • Next, multiply the bottom numbers (denominators):
    \( 4 \times 6 = 24 \).
  • This gives you the fraction \( \frac{1}{24} \).
A handy tip is to check if any cross-canceling can happen before multiplying, but in our example, it wasn鈥檛 necessary since both numbers are already simplified. Multiplying fractions is all about following these simple steps.
Squaring Fractions
Squaring a fraction might sound tricky, but it鈥檚 actually as simple as multiplying it by itself. Here's how it works in detail:
  • Take your fraction, for example, \( \left(-\frac{5}{6}\right) \).
  • Square the numerator:
    \((-5)^2 = 25\). Don鈥檛 forget that squaring a negative results in a positive number.
  • Then square the denominator:
    \(6^2 = 36\).
  • Place the results into a fraction:
    \( \frac{25}{36} \).
Squaring fractions follows a familiar pattern like squaring whole numbers. Ensuring that you remember both parts, the numerator and the denominator, are squared separately is key.
Adding Fractions
Adding fractions comes with the unique challenge of finding a common denominator. Here鈥檚 how you can tackle it step by step:
  • Look at the fractions you need to add: \( \frac{1}{24} \) and \( \frac{25}{36} \).
  • Find the least common denominator that both fractions can share. For 24 and 36, it鈥檚 72.
  • Convert each fraction to this new denominator:
    \( \frac{1}{24} = \frac{3}{72} \) by multiplying both the numerator and the denominator by 3.
    \( \frac{25}{36} = \frac{50}{72} \) by multiplying both the numerator and the denominator by 2.
  • Now, add the fractions:
    \( \frac{3}{72} + \frac{50}{72} = \frac{53}{72} \).
After getting a common denominator, it's just like adding simple numbers on top while keeping the denominator the same. Finally, check if you can simplify.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$\frac{9}{4} x+\frac{4}{3}=-\frac{1}{6}$$

Solve the equation and simplify your answer. $$-\frac{5}{2} x+\frac{9}{5}=\frac{5}{8}$$

Solve the equation and simplify your answer. $$2 x-3=6 x+7$$

Solve the equation and simplify your answer. $$6 x+9=-6 x$$

Solve the equation and simplify your answer. $$x+\frac{8}{9}=\frac{2}{3}$$
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