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Chapter 4: Problem 9

Calculate: a. \(\left(\frac{1}{100}\right)^{1 / 2}\) b. \(25^{-1 / 2}\) c. \(\left(\frac{9}{16}\right)^{-1 / 2}\) d. \(\left(\frac{1}{1000}\right)^{1 / 3}\)

Short Answer

Expert verified
a. \( \frac{1}{10} \) \ewline b. \( \frac{1}{5} \) \ewline c. \( \frac{4}{3} \) \ewline d. \( \frac{1}{10} \)

Step by step solution

01

Calculate \left( \frac{1}{100} \right)^{1/2}

The exponent \( \frac{1}{2} \) represents the square root. So \( \left( \frac{1}{100} \right)^{1/2} = \sqrt{ \frac{1}{100} } = \frac{1}{10} \).
02

Calculate \(25^{-1/2}\)

The exponent \( -1/2 \) represents the reciprocal of the square root. \( 25^{-1/2} = \left( 25 \right)^{-1/2} = \frac{1}{\sqrt{25}} = \frac{1}{5} \).
03

Calculate \left( \frac{9}{16} \right)^{-1/2}

The exponent \( -1/2 \) means we first take the reciprocal and then the square root. \( \left( \frac{9}{16} \right)^{-1/2} = \left( \frac{16}{9} \right)^{1/2} = \sqrt{ \frac{16}{9} } = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \).
04

Calculate \left( \frac{1}{1000} \right)^{1/3}

The exponent \(1/3\) represents the cube root. So \(\left( \frac{1}{1000} \right)^{1/3} = \sqrt[3]{\frac{1}{1000}} = \frac{1}{\sqrt[3]{1000}} = \frac{1}{10} \).

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Key Concepts

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

Fractional Exponents
Fractional exponents are a way to express roots using exponents. For example, the exponent \(\frac{1}{2}\) represents a square root. When you see an expression like \(a^{\frac{1}{2}}\), it means \(\sqrt{a}\). Similarly, \(a^{\frac{1}{3}}\) means the cube root, written as \(\sqrt[3]{a}\). This concept simplifies the notation and calculation of roots. Always remember that the denominator in the fraction tells us which root to take, and the numerator is the power to raise the number by.

For example:
- \(8^{\frac{1}{3}} = \sqrt[3]{8} = 2\)
- \(16^{\frac{1}{4}} = \sqrt[4]{16} = 2\)
Reciprocals
A reciprocal of a number is one divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). When using exponents, a negative exponent indicates a reciprocal. For example, \(a^{-n}\) means \(\frac{1}{a^n}\). This simplifies calculations by transforming divisions into multiplications. In our solved exercise, \(25^{-\frac{1}{2}} = \frac{1}{\sqrt{25}} = \frac{1}{5}\). Notice how the negative exponent made us take the reciprocal of the square root.

Some more examples:
- \(10^{-1} = \frac{1}{10}\)
- \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
Square Roots
A square root of a number is another number that, when multiplied by itself, gives the original number. It's denoted as \(\sqrt{a}\). For fractional exponents, the square root is represented by \(a^{\frac{1}{2}}\). For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\). In our exercise, \(\left(\frac{1}{100}\right)^{\frac{1}{2}} = \sqrt{\frac{1}{100}} = \frac{1}{10}\). This transformation makes it easier to handle root calculations.

More examples:
- \(\sqrt{9} = 3\)
- \(\sqrt{49} = 7\)
Cube Roots
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. It's written as \(\sqrt[3]{a}\). With exponents, cube roots are shown as \(a^{\frac{1}{3}}\). For example, \(\sqrt[3]{27} = 3\) because \(3 \times 3 \times 3 = 27\). In the provided exercise, \(\left(\frac{1}{1000}\right)^{\frac{1}{3}} = \sqrt[3]{\frac{1}{1000}} = \frac{1}{10}\). This relationship simplifies finding roots in more complex expressions.

Additional examples:
- \(\sqrt[3]{8} = 2\)
- \(\sqrt[3]{64} = 4\)

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Most popular questions from this chapter

The concentration of hydrogen ions in a water solution typically ranges from \(10 \mathrm{M}\) to \(10^{-15} \mathrm{M}\). (One \(\mathrm{M}\) equals \(6.02 \cdot 10^{23}\) particles, such as atoms, ions, molecules, etc., per liter or 1 mole per liter.) Because of this wide range, chemists use a logarithmic scale, called the pH scale, to measure the concentration (see Exercise 12 of Section 4.6 ). The formal definition of \(\mathrm{pH}\) is \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where \(\left[\mathrm{H}^{+}\right]\) denotes the concentration of hydrogen ions. Chemists use the symbol \(\mathrm{H}^{+}\) for hydrogen ions, and the brackets [ ] mean "the concentration of." a. Pure water at \(25^{\circ} \mathrm{C}\) has a hydrogen ion concentration of \(10^{-7} \mathrm{M}\). What is the \(\mathrm{pH} ?\) b. In orange juice, \(\left[\mathrm{H}^{+}\right] \approx 1.4 \cdot 10^{-3} \mathrm{M}\). What is the \(\mathrm{pH}\) ? c. Household ammonia has a pH of about \(11.5 .\) What is its \(\left[\mathrm{H}^{+}\right] ?\) d. Does a higher pH indicate a lower or a higher concentration of hydrogen ions? e. A solution with a \(\mathrm{pH}>7\) is called basic, one with a \(\mathrm{pH}=7\) is called neutral, and one with a \(\mathrm{pH}<7\) is called acidic. Identify pure water, orange juice, and household ammonia as either acidic, neutral, or basic. Then plot their positions on the accompanying scale, which shows both the \(\mathrm{pH}\) and the hydrogen ion concentration.

Evaluate the following without a calculator. a. Find the following values: i. \(\log 100\) ii. log 1000 iii. \(\log 10,000,000\) What is happening to the values of \(\log x\) as \(x\) gets larger? b. Find the following values: i. \(\log 0.1\) ii. \(\log 0.001\) iii. \(\log 0.00001\) What is happening to the values of \(\log x\) as \(x\) gets closer to \(0 ?\) c. What is \(\log 0 ?\) d. What is \(\log (-10) ?\) What do you know about \(\log x\) when \(x\) is any negative number?

Evaluate without a calculator: a. \(\sqrt{10,000}\) b. \(\sqrt{-25}\) c. \(625^{1 / 2}\) d. \(100^{1 / 2}\) e. \(\left(\frac{1}{9}\right)^{1 / 2}\) f. \(\left(\frac{625}{100}\right)^{1 / 2}\)

Without using a calculator, find two consecutive integers such that one is smaller and one is larger than each of the following (for example, \(3<\sqrt{11}<4\) ). Show your reasoning. a. \(\sqrt{13}\) b. \(\sqrt{22}\) c. \(\sqrt{40}\)

Rewrite in an equivalent form using logarithms: a. \(10^{4}=10.000\) b. \(10^{-2}=0.01\) c. \(10^{0}=1\) d. \(10^{-5}=0.00001\)
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