Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 8^{-\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplest form of \(8^{-\frac{1}{3}}\) is \(\frac{1}{2}\).

Step 1: Convert to Fractional Exponent

The given expression has a negative exponent: \(8^{-\frac{1}{3}}\). A negative exponent indicates the reciprocal of the base raised to the positive equivalent of the exponent. Thus, \(8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}}\).

Step 2: Find the Cube Root

Now, resolve the fractional exponent \(8^{\frac{1}{3}}\) by taking the cube root of 8. The cube root of 8 is 2, i.e., \(8^{\frac{1}{3}} = 2\).

Step 3: Calculate the Reciprocal

Place your result from Step 2 as the denominator in the reciprocal to get the simplest form of the power. Therefore, \(8^{-\frac{1}{3}} = \frac{1}{2}\).

Key Concepts

Negative Exponents

When faced with a negative exponent, such as in the expression \(8^{-\frac{1}{3}}\), it's essential to understand that a negative exponent signifies the reciprocal of the base raised to a positive exponent.- Let's break this down: - \(a^{-b} = \frac{1}{a^b}\)This means when you see \(8^{-\frac{1}{3}}\), you should think about flipping the base to become a fraction. So, it gets transformed into \(\frac{1}{8^{\frac{1}{3}}}\).
The negative sign has served its purpose by changing the expression to a fraction. This technique is crucial because it simplifies complex expressions into more manageable ones. Understanding negative exponents can significantly aid in dealing with algebraic expressions and equations efficiently.

Fractional Exponents

Fractional exponents might initially look intimidating, but they are merely another way to express roots and powers together. The general form of a fractional exponent is \(a^{\frac{m}{n}}\), where:- \(m\) is the power (you raise the number to this power)- \(n\) is the root (you take the nth root of the number)For example, in \(8^{\frac{1}{3}}\), the exponent's denominator, 3, tells us to take the cube root of 8. So \(8^{\frac{1}{3}}\) becomes \(\sqrt[3]{8}\), which is equal to 2. The numerator of the fractional exponent, which was 1 in this case, translates to the power part (raising to a power of 1 does not change the value).
This understanding lets you not only transform roots into powers but also allows you to simplify expressions that involve both roots and powers seamlessly.

Cube Roots

Cube roots are a specific type of root where you find the value that, when multiplied by itself three times, gives the original number. In mathematical terms, the cube root of \(x\) is expressed as \(\sqrt[3]{x}\) or equivalently, \(x^{\frac{1}{3}}\).Finding cube roots is simpler with whole numbers that are perfect cubes, like 8. The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). For numbers that are not perfect cubes, finding cube roots can involve approximation or more complex calculations. Cube roots are significant in various fields like geometry and physics due to their association with volume and three-dimensional space.
The ability to break down a number using cube roots and relate them to fractional exponents enhances your mathematical problem-solving toolkit.