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Chapter 9: Problem 148

Explain how would you evaluate an expression with a mixed-number exponent. For example, what is \(8^{1 \frac{1}{3}} ?\) What is \(25^{2 \frac{1}{2}} ?\)

Short Answer

Expert verified
\(8^{1 \frac{1}{3}} = 16\); \(25^{2 \frac{1}{2}} = 3125\).

Step by step solution

01

Understanding Mixed-Number Exponents

A mixed-number exponent like \( 1 \frac{1}{3} \) represents the sum of an integer and a fraction. For \( 1 \frac{1}{3} \), it鈥檚 equivalent to \( 1 + \frac{1}{3} \). Therefore, \( 8^{1 \frac{1}{3}} \) can be rewritten as \( 8^{1 + \frac{1}{3}} \), which is the same as \( 8^1 \times 8^{\frac{1}{3}} \).
02

Simplifying Integer Exponent

First, simplify the integer exponent part. \( 8^1 \) is simply \( 8 \). So the expression becomes \( 8 \times 8^{\frac{1}{3}} \).
03

Evaluating the Fractional Exponent

The fractional exponent \( \frac{1}{3} \) indicates a cube root. Therefore, \( 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 \). So the expression \( 8 \times 8^{\frac{1}{3}} \) becomes \( 8 \times 2 = 16 \).
04

Evaluating Second Expression

For \( 25^{2 \frac{1}{2}} \), rewrite it as \( 25^2 \times 25^{\frac{1}{2}} \). \( 25^2 = 625 \) and \( 25^{\frac{1}{2}} = \sqrt{25} = 5 \).
05

Final Calculation

Multiply the results from previous calculations to find the final value. \( 625 \times 5 = 3125 \). Thus, \( 25^{2 \frac{1}{2}} \) equals \( 3125 \).

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Key Concepts

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

Mixed-Number Exponents
Working with mixed-number exponents might initially seem tricky, but they can be quite simple once you break them down. A mixed-number exponent combines an integer and a fraction. For instance, in the expression \( 8^{1 \frac{1}{3}} \), the exponent \( 1 \frac{1}{3} \) can be seen as \( 1 + \frac{1}{3} \).
To simplify this, you want to split the exponent into separate parts: the integer part (1) and the fractional part (\( \frac{1}{3} \)).
  • So, \( 8^{1 \frac{1}{3}} \) can be rewritten as \( 8^1 \times 8^{\frac{1}{3}} \).
  • Then you solve \( 8^1 \) and \( 8^{\frac{1}{3}} \) separately.
This makes it easier to handle because you've reduced the problem to more manageable pieces.
Fractional Exponents
Fractional exponents represent roots. They tell us to take specific roots of a number rather than multiplying the base by itself multiple times.
When you see an exponent such as \( a^{\frac{1}{n}} \), it means you're finding the \( n \)-th root of \( a \).
  • For example, \( 8^{\frac{1}{3}} \) means you're finding the cube root of 8, which is 2.
  • Similarly, \( 25^{\frac{1}{2}} \) means you're finding the square root of 25, which is 5.
Converting fractional exponents to root form can make calculations more straightforward, especially when using calculators or simplifying manually.
Cube Root
The cube root of a number is what we multiply by itself three times to get that number. It's denoted using the radical symbol with a little 3 above it, like this: \( \sqrt[3]{x} \).
Understanding cube roots is essential when dealing with cubic equations or simplifying expressions with fractional exponents of \( \frac{1}{3} \).
  • For instance, \( \sqrt[3]{8} = 2 \), because \( 2 \times 2 \times 2 = 8 \).
  • Another example: \( \sqrt[3]{27} = 3 \), since \( 3 \times 3 \times 3 = 27 \).
When tackling mixed-number exponents like \( 8^{1\frac{1}{3}} \), identifying the cube root part helps simplify the expression quickly.
Square Root
The square root function simplifies expressions by identifying a number that, when multiplied by itself, equals the original number. It's symbolized with the radical sign: \(\sqrt{x}\).
Square roots are easy to spot in fractional exponents like \( \frac{1}{2} \).
  • \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).
  • \( \sqrt{4} = 2 \) since \( 2 \times 2 = 4 \).
This concept often appears in equations and is especially useful for simplifying expressions involving fractional exponents like \( 25^{1\frac{1}{2}} \). Recognizing square roots will make solving such problems much easier.

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Most popular questions from this chapter

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[9]{\sqrt{c}} $$

Use a calculator to solve each problem. Round answers to the nearest tenth. Shoelaces. The formula \(S=2[H+L+(p-1) \sqrt{H^{2}+V^{2}}]\) can be used to calculate the correct shoelace length for the criss-cross lacing pattern shown in the illustration, where \(p\) represents the number of pairs of eyelets. Find the correct shoelace length if \(H\) (horizontal distance) \(=50 \mathrm{mm}, L\) (length of end) \(=250 \mathrm{mm},\) and \(V(\) vertical distance \()=20 \mathrm{mm} .\) Round to the nearest tenth.

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[10]{x^{2} y^{2}} $$

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Use a calculator to solve each problem. Round answers to the nearest tenth. The Grand Canyon. The time \(t\) (in seconds) that it takes for an object to fall a distance of \(s\) feet is given by the formula \(t=\frac{\sqrt{s}}{4}\) In some places, the Grand Canyon is one mile ( \(5,280\) feet) deep. How long would it take a stone dropped over the edge of the canyon to hit bottom?
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