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Chapter 7: Problem 4

Use radical notation to write each expression. Simplify if possible. $$ 8^{1 / 3} $$

Short Answer

Expert verified
\(8^{1/3} = \sqrt[3]{8} = 2\)

Step by step solution

01

Understanding the Radical Notation

The exercise asks to express an exponent in radical form. In mathematics, a fractional exponent, like \(a^{1/n}\), can be rewritten using a radical. Specifically, \(a^{1/n}\) can be expressed as \(\sqrt[n]{a}\), which is the \(n\)-th root of \(a\). This is called converting from exponential to radical notation.
02

Convert the Exponent to Radical Notation

Given the expression \(8^{1/3}\), we identify that the exponent is 1/3. This means we need to find the cube root of 8. Using the rule from Step 1, we rewrite the expression as a radical: \(8^{1/3} = \sqrt[3]{8}\).
03

Simplify the Radical

Now we simplify \(\sqrt[3]{8}\). We seek a number which, when raised to the power of 3, equals 8. That number is 2, since \(2^3 = 8\). Therefore, \(\sqrt[3]{8} = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractional Exponents
Fractional exponents might seem confusing at first, but they actually offer a compact way to represent roots. When you see something like \(a^{m/n}\), you are looking at both an exponent and a root in disguise. The numerator \(m\) is your usual exponent, but the denominator \(n\) tells you the root. So \(a^{1/3}\) would mean "find the cube root of \(a\)," and \(a^{2/3}\) would translate to "square \(a\), then find the cube root." With fractional exponents, we:
  • Convert the denominator into the root (e.g., \(x^{1/2}\) becomes \(\sqrt{x}\))
  • The numerator stays as the power (e.g., \(x^{3/2}\) becomes \((\sqrt{x})^3\))
Breaking down complex terms into simple roots and powers can make calculations more straightforward, especially when trying to simplify expressions.
Cube Root
The cube root operates in a similar way to the square root but focuses on finding the number which, when multiplied by itself three times, gives your original value. When dealing with \(\sqrt[3]{x}\) or \(x^{1/3}\), you're effectively asking, "What number cubed equals \(x\)?" Let's look at the properties:
  • Finding a cube root is the opposite of cubing a number (if \(y^3 = x\), then \(\sqrt[3]{x} = y\)).
  • Cube roots can handle negative numbers because multiplying three negative values results in a negative value (e.g., \((-2)^3 = -8\) implies \(\sqrt[3]{-8} = -2\)).
Cube roots are not just limited to perfect cubes but can often be approximated when perfect solutions aren't possible mathematically or manually.
Simplifying Radicals
Simplifying radicals is all about making expressions as straightforward as possible. When you have complex expressions, simplifying makes them easier for problem-solving and understanding. Here's how to simplify:
  • Look for perfect powers within the radical (e.g., \(\sqrt[3]{8}\) simplifies to 2 because \(2^3 = 8\)).
  • If the number isn't a perfect root, see if you can break it into components that are (e.g., \(\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2} \)).
  • Simplifying roots like these keeps your math neat and error-free when working with complex problems.
Remember, simplifying radicals not only helps in computations but also enhances clarity when reading and interpreting mathematical expressions.

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Most popular questions from this chapter

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}} $$

Use a calculator to approximate each square root to 3 decimal places. $$ \sqrt{56} $$

The tallest structure in the United States is a TV tower in Blanchard, North Dakota. Its height is 2063 feet. A 2382 -foot length of wire is to be used as a guy wire attached to the top of the tower. Approximate to the nearest foot how far from the base of the tower the guy wire must be anchored.

Rationalize each denominator. See Example 4. $$ \frac{6}{2-\sqrt{7}} $$
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