Problem 51 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$3\left(\frac{1}{2}\right)^{4}-2\left(\frac{1}{2}\right)^{3}+5\left(\frac{1}{2}\right)^{2}-4\left(\frac{1}{2}\right)+1$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{16} \).

Step 1: Calculate Powers of the Fraction

Begin by evaluating the powers of \( \left(\frac{1}{2}\right) \) in the expression. - \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) - \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \) - \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \)

Step 2: Multiply by Coefficients

Multiply each result from Step 1 by its respective coefficient in the expression. - \( 3\left(\frac{1}{2}\right)^4 = 3 \times \frac{1}{16} = \frac{3}{16} \)- \( -2\left(\frac{1}{2}\right)^3 = -2 \times \frac{1}{8} = -\frac{2}{8} = -\frac{1}{4} \)- \( 5\left(\frac{1}{2}\right)^2 = 5 \times \frac{1}{4} = \frac{5}{4} \)- \( -4\left(\frac{1}{2}\right) = -4 \times \frac{1}{2} = -2 \)

Step 3: Simplify the Combined Expression

Combine all terms from Step 2, including the constant, into a single expression: \[ \frac{3}{16} - \frac{1}{4} + \frac{5}{4} - 2 + 1 \]

Step 4: Find a Common Denominator

Convert all terms into fractions with a common denominator (16): - \( \frac{1}{4} = \frac{4}{16} \) - \( \frac{5}{4} = \frac{20}{16} \)- \( -2 = \frac{-32}{16} \)- \( 1 = \frac{16}{16} \)

Step 5: Add and Subtract the Fractions

Combine all fractions from Step 3 using the common denominator:\[ \frac{3}{16} - \frac{4}{16} + \frac{20}{16} - \frac{32}{16} + \frac{16}{16} \]Calculate step by step: - \( \frac{3}{16} - \frac{4}{16} = -\frac{1}{16} \)- \( -\frac{1}{16} + \frac{20}{16} = \frac{19}{16} \) - \( \frac{19}{16} - \frac{32}{16} = -\frac{13}{16} \)- \( -\frac{13}{16} + \frac{16}{16} = \frac{3}{16} \)

Step 6: Conclusion of Simplification

After adding and subtracting all terms, the simplified expression is \( \frac{3}{16} \).

Key Concepts

Fractional Exponents

When dealing with fractional exponents, you are often handling expressions that look like this: \( rac{a}{b} \), where "a" is the exponent and "b" is the root.
For example, \( rac{1}{2} \) raised to a power is just one part of the sequence of operations involving these fractional exponents.
Working with these requires calculating each power independently before any other operation like multiplication or addition.In the given exercise, we find that calculating \( \left(\frac{1}{2}\right)^4 \), \( \left(\frac{1}{2}\right)^3 \), and \( \left(\frac{1}{2}\right)^2 \) yields results using simple multiplication of the base \( \frac{1}{2} \), repeated as many times as the exponent states.

• \( \left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
• \( \left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)
• \( \left(\frac{1}{2}\right)^4 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16} \)

Understanding these steps makes it simpler to work through more complex expressions.

Common Denominators

Combining fractions directly is not possible unless they have a common denominator.
This commonality allows us to directly add or subtract fractions by focusing just on the numerators, keeping the denominator constant.In our exercise, once we calculated separate fractions, we needed to convert them into a consistent format or common denominator.
For fractions like \( \frac{1}{4} \), \( \frac{5}{4} \), \( -2 \), and \( 1 \), the common denominator was chosen to be 16, the largest denominator present.
This is achieved by multiplying the top (numerator) and the bottom (denominator) of each fraction so that all the denominators match this common one:

• Convert \( \frac{1}{4} \) into \( \frac{4}{16} \) by multiplying both parts by 4.
• Convert \( \frac{5}{4} \) into \( \frac{20}{16} \) by multiplying both parts by 4.
• Change \( -2 \) to \( \frac{-32}{16} \) as \( -2 = \frac{-32}{16} \) naturally.
• Modify \( 1 \) to \( \frac{16}{16} \) since \( 1 = \frac{16}{16} \).

With the same denominator of 16, the fractions can easily be managed together.

Addition and Subtraction of Fractions

Once we've established a common denominator, the next step is to perform the addition or subtraction of the fractions.
This process becomes straightforward because we are primarily working with the numerators while the denominator remains unchanged.In this particular exercise, fractions are added and subtracted following a sequential process:

• Start with \( \frac{3}{16} - \frac{4}{16} \) to simplify to \( -\frac{1}{16} \).
• Then \( -\frac{1}{16} + \frac{20}{16} \) results in \( \frac{19}{16} \).
• Then \( \frac{19}{16} - \frac{32}{16} \) simplifies to \( -\frac{13}{16} \).
• Finally, \( -\frac{13}{16} + \frac{16}{16} \) yields \( \frac{3}{16} \).

This sequence showcases not just the clarity of a common denominator but also provides an efficient way to handle mixed operations involving fractions.