Problem 4 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ 5\left(\frac{1}{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression is 40.

Step 1: Understand the Negative Exponent

The expression \( \left( \frac{1}{2} \right)^{-3} \) involves a negative exponent. A negative exponent means that the base (in this case, \( \frac{1}{2} \)) should be reciprocal and the exponent then becomes positive. Reciprocating \( \frac{1}{2} \) gives you \( 2 \), because the reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \) or simply \( 2 \).

Step 2: Apply the Positive Exponent

Now that we have \( 2 \) as the base, raise it to the positive power of 3. Thus, we calculate \( 2^3 \). This means multiplying 2 by itself three times: \[2 \times 2 \times 2 = 8\].

Step 3: Multiply by the Coefficient

Multiply the simplified value from Step 2 by the coefficient outside the parentheses, which is 5. So you have: \[5 \times 8 = 40\].

Step 4: Final Result

Combine all steps to get the final result of the expression. The expression \( 5 \left( \frac{1}{2} \right)^{-3} \) simplifies to 40.

Key Concepts

Negative Exponent

Negative exponents can often be confusing, but they follow a straightforward rule. The rule tells us that when you encounter a negative exponent, such as in the expression \( \left( \frac{1}{2} \right)^{-3} \), it indicates that you need to take the reciprocal of the base.

Here's what happens when you deal with a negative exponent:

• The base is \( \frac{1}{2} \), and the exponent is \(-3\).
• Because of the negative, you find the reciprocal. The reciprocal of \( \frac{1}{2} \) is 2, because \( \frac{2}{1} = 2 \).
• After taking the reciprocal, the exponent becomes positive.

This turns the expression into \( 2^3 \), which is much more straightforward to solve. Recognizing and understanding this rule will make solving expressions with negative exponents much simpler.

Reciprocal

Understanding reciprocals is vital when dealing with expressions that involve negative exponents. A reciprocal is essentially flipping a fraction.

For example:

• If your original number is \( \frac{1}{2} \), its reciprocal will be \( \frac{2}{1} \) or simply 2.
• Whenever you see a fraction, remember that the reciprocal just swaps the numerator and the denominator.

Once you have the reciprocal, you can switch the negative exponent to a positive one. In our example, changing \( \left( \frac{1}{2} \right)^{-3} \) to a positive exponent involves first finding the reciprocal, which simplifies your calculation to \( 2^3 \). This step is crucial as it lays the foundation for simplifying expressions with negative exponents effectively.

Simplifying Expressions

Simplifying expressions makes solving math problems much easier and clearer. The key is to break down the expression step by step. For the given exercise, once the negative exponent has been addressed and the reciprocal found, you can move into straightforward calculations.

Consider these steps:

• First, identify the need to convert the negative exponent into a positive one, using its reciprocal.
• Then, solve the exponentiation by multiplying the base by itself as many times as indicated by the exponent.
• Finally, multiply by any outside coefficients to get the simplified expression.

In our specific case, once we calculated \( 2^3 = 8 \), it was just a matter of multiplying that result by 5, leading us to the final answer of 40. By following these steps systematically, you can simplify even complex-looking expressions efficiently, enhancing your understanding and confidence in handling algebraic expressions.