Problem 37 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 8^{-2 / 3} $$

Step-by-Step Solution

Answer

8^{-2/3} = \(\frac{1}{4}\).

Step 1: Understand the Negative Exponent

The negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, the expression \(8^{-2/3}\) can be rewritten as \(\frac{1}{8^{2/3}}\).

Step 2: Simplify the Fractional Exponent

The fractional exponent \(\frac{2}{3}\) can be interpreted as a root and a power. Specifically, it means we first take the cube root of 8 and then square it, or vice versa: \(8^{2/3} = (8^{1/3})^2 = (2)^2 = 4\).

Step 3: Evaluate the Expression with Reciprocal

Now, apply the reciprocal operation from Step 1. Instead of \(8^{2/3}\), which we found to be 4, we calculate \(\frac{1}{4}\) since it was \((8^{2/3})\) negative. So, \(8^{-2/3} = \frac{1}{4}\).

Key Concepts

Negative Exponents

When you encounter a negative exponent, it's an indication to take the reciprocal (or flip) of the base raised to the corresponding positive exponent. For instance, in the expression \(8^{-2/3}\), the negative exponent "points" out that we need to take \(1/8\) raised to the power of \(2/3\). This turns the expression into \(\frac{1}{8^{2/3}}\). In simpler terms, a negative exponent tells us how many times to divide 1 by the base, instead of multiplying it. This flips the power from the numerator to the denominator in a fraction.

• Example: \(x^{-a} = \frac{1}{x^a}\)
• Think of it as the opposite of multiplying: it's repeatedly taking the reciprocal.

Fractional Exponents

Fractional exponents might look tricky, but they represent a combination of roots and powers. The expression \(8^{2/3}\) can be broken down to mean the cube root of 8, all raised to the power of 2. More generally, the denominator of the fraction (3 in this case) directs us to take that root of the base, while the numerator (2 here) indicates how many times to multiply that result by itself. Therefore, \(8^{1/3}\) gives us 2, since 2 cubed equals 8. Then, we square that result: \((2)^2 = 4\).

• General Rule: \(b^{m/n} = (b^{1/n})^m\)
• The denominator "n" in the exponent signifies the nth root, while the numerator "m" indicates the power.

Reciprocal of Powers

The concept of the reciprocal is essential when dealing with negative exponents. After simplifying a fractional exponent, we use the idea of reciprocal to complete the evaluation. If given \(8^{-2/3}\), we first find \(8^{2/3}\), which equals 4, by interpreting the fractional exponent. Then, we apply the negative exponent by taking the reciprocal of the result. Hence, we take \(\frac{1}{4}\). It's a handy way to understand inverse operations in power calculations.

In essence, the reciprocal operation flips the direction of multiplication, transforming power operations into division. Whenever you see a negative exponent, always remember it's telling you to "flip" the process by turning the answer upside down, quite literally!

• Reciprocal: Turning \(a^b\) into \(\frac{1}{a^b}\)
• Helps in simplifying expressions involving negative exponents effectively.