Problem 7 - Algebra for College Students

Question

Simplify each numerical expression. \(-\left(\frac{1}{3}\right)^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is -27.

Step 1: Identify the Negative Exponent

The problem involves simplifying \(-\left(\frac{1}{3}\right)^{-3}\). A negative exponent indicates the reciprocal of the base raised to the positive exponent. Here, the base is \(\frac{1}{3}\) and the exponent is \(-3\). The negative sign in front of the expression doesn't affect this specific rule for negative exponents.

Step 2: Apply the Rule of Negative Exponents

The rule for negative exponents states that \(x^{-n} = \frac{1}{x^n}\). Applying this rule, \(\left(\frac{1}{3}\right)^{-3}\) becomes \(\left(\frac{3}{1}\right)^{3}\), which simplifies to \(3^3\). So, the expression becomes \(-3^3\).

Step 3: Calculate the Positive Exponent

Now, we calculate \(3^3\). To do this, multiply 3 by itself three times: \(3 \times 3 \times 3 = 27\).

Step 4: Apply the Negative Sign

Finally, apply the negative sign from the beginning of the expression. Hence, \(-3^3 = -27\).

Key Concepts

Reciprocal

When dealing with negative exponents, it's important to understand the concept of a reciprocal. A reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 3 is \( \frac{1}{3} \). This concept is incredibly useful when trying to simplify expressions with negative exponents.

• When you see a negative exponent, you can think of it as an indicator to take the reciprocal of the base.
• Negative exponents tell us to "flip" the base, switch the numerator and denominator before applying the exponent.

So, for any expression like \( \left( \frac{1}{3} \right)^{-3} \), the negative exponent flips the base fraction to \( \left( \frac{3}{1} \right)^{3} \), making it a lot easier to work with.

Exponential Notation

Exponential notation is a convenient way to express repeated multiplication of the same number. If you have a base \(a\) and an exponent \(n\), then \(a^n\) signifies that you multiply \(a\) by itself \(n\) times. For example, \(3^3\) equals \(3 \times 3 \times 3 = 27\).

• The exponent itself never affects the sign in front of the expression, unless specifically paired with a base and parentheses.
• Exponential notation simplifies complex calculations and helps convey mathematical ideas cleanly.

This notation becomes very helpful when dealing with expressions like \(-\left( \frac{3}{1} \right)^{3}\), where you want to quickly simplify to \(-27\). Notice how you can easily see the steps to resolve the expression into its final value.

Simplifying Expressions

Simplifying expressions in mathematics means reducing them to their simplest form. It involves following a series of mathematical rules and operations to ensure clarity and precision in the result.

• For expressions like \(\left( \frac{1}{3} \right)^{-3}\), simplifying involves recognizing the negative exponent, finding the reciprocal, and then calculating the positive power.
• By applying the rules of exponents and mathematical operations systematically, the expression \(-3^3\) becomes \(-27\).

Always remember to keep track of negative signs, as they impact the final result. Simplifying expressions not only helps in calculations but also enhances overall understanding of algebraic concepts.