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Chapter 4: Problem 8

Calculate the following: a. \(4^{1 / 2}\) b. \(-4^{1 / 2}\) c. \(27^{1 / 3}\) d. \(-27^{1 / 3}\) e. \(8^{2 / 3}\) f. \(-8^{2 / 3}\) g. \(16^{1 / 4}\) h. \(16^{3 / 4}\)

Short Answer

Expert verified
a. 2, b. -2, c. 3, d. -3, e. 4, f. -4, g. 2, h. 8.

Step by step solution

01

Calculate a. \(4^{1/2}\)

To find the value of \(4^{1/2}\), recognize that raising a number to the power of \(1/2\) is equivalent to taking the square root of that number. Thus, \(4^{1/2} = \sqrt{4} = 2\).
02

Calculate b. \(-4^{1/2}\)

The value of \(-4^{1/2}\) implies the negative of the square root of 4 as calculated in the previous step. Therefore, \(-4^{1/2} = -\sqrt{4} = -2\).
03

Calculate c. \(27^{1/3}\)

Raising a number to the power of \(1/3\) is equivalent to taking the cube root of that number. Hence, \(27^{1/3} = \sqrt[3]{27} = 3\).
04

Calculate d. \(-27^{1/3}\)

The value of \(-27^{1/3}\) is the negative of the cube root of 27. Consequently, \(-27^{1/3} = -\sqrt[3]{27} = -3\).
05

Calculate e. \(8^{2/3}\)

To calculate \(8^{2/3}\), first determine the cube root of 8, then square that result. The cube root of 8 is 2, and squaring 2 gives: \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\).
06

Calculate f. \(-8^{2/3}\)

The value of \(-8^{2/3}\) is the negative of \(8^{2/3}\) as calculated in the previous step. Therefore, \(-8^{2/3} = -4\).
07

Calculate g. \(16^{1/4}\)

The value of \(16^{1/4}\) is equivalent to taking the fourth root of 16. Hence, \(16^{1/4} = \sqrt[4]{16} = 2\).
08

Calculate h. \(16^{3/4}\)

To find \(16^{3/4}\), first determine the fourth root of 16, then raise that result to the power of 3. The fourth root of 16 is 2, and raising 2 to the power of 3 gives: \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\).

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Key Concepts

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

Rational Exponents
Rational exponents might sound complicated, but they鈥檙e not too hard to understand. When we see something like \(4^{1/2}\), it's the same as taking the square root of 4. Here鈥檚 a simple way to think about it: the denominator of the exponent (in this case, 2) tells us what kind of root to take. So, \(4^{1/2} = \sqrt{4} = 2\).
Another example is \(27^{1/3}\). Here, the denominator is 3, so we take the cube root: \(27^{1/3} = \sqrt[3]{27} = 3\).
If you have an exponent like \(8^{2/3}\), you take the cube root first (which is 2), then raise it to the power of 2, so \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\).
Roots
Roots are just another way of thinking about exponents. For example, the square root of a number is the same as raising that number to the power of \(1/2\), like \(4^{1/2}\).
The cube root is the same as raising a number to the power of \(1/3\), like \(27^{1/3}\).
Sometimes you鈥檒l see other roots, like the fourth root. This is written as \(16^{1/4}\) and means we find a number which, when raised to the power of 4, gives us 16. Here it鈥檚 2 because \(2^4 = 16\).
Negative Exponents
Negative exponents can be intimidating, but they follow simple rules. If you see something like \(-4^{1/2}\), it's the same as taking the square root of 4 and then making the result negative.
So, \(-4^{1/2} = -\sqrt{4} = -2\). Similarly, \(-27^{1/3}\) means taking the cube root of 27 and making it negative: \(-27^{1/3} = -\sqrt[3]{27} = -3\).
For something like \(-8^{2/3}\), you first find \(8^{2/3}\) and then make the result negative, like \(-4\).
Simplifying Expressions
Simplifying expressions with exponents often requires breaking the problem into smaller parts. For example, to simplify \(16^{3/4}\), you first find \(16^{1/4}\), which is 2 because \(\sqrt[4]{16} = 2\).
Then raise 2 to the power of 3 to get 8: \(16^{3/4} = 2^3 = 8\).
Step-by-step solutions help, like computing \(8^{2/3}\) by first finding \(8^{1/3}\), which is 2, and then squaring the result: \(2^2 = 4\).

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Most popular questions from this chapter

In the United States, land is measured in acres and one acre is 43,560 sq \(\mathrm{ft}\) a. If you buy a one-acre lot that is in the shape of a square, what would be the length of each side in feet? b. A newspaper advertisement states that all lots in a new housing development will be a minimum of one and a half acres. Assuming the lot is rectangular and has \(150 \mathrm{ft}\) of frontage, how deep will the minimal-size lot be? If the new home owner wants to fence in the lot, how many yards of fencing would be needed? c. The metric unit for measuring land is the square hectometer. (A hectometer is a length of 100 meters.) Find the size of a one-acre lot if it were measured in square hectometers. d. A hectare is 100 acres. How many one-acre lots can fit in a square mile? How many hectares is that?

Write each of the following in scientific notation: a. \(725 \cdot 10^{23}\) b. \(725 \cdot 10^{-23}\) c. \(\frac{1}{725 \cdot 10^{23}}\) d. \(-725 \cdot 10^{23}\) e. \(-725 \cdot 10^{-23}\)

a. A roll of aluminum foil claims to be \(50 \mathrm{sq} \mathrm{ft}\) or \(4.65 \mathrm{~m}^{2}\). Show the conversion factors that would verify that these two measurements are equivalent. b. One \(\mathrm{cm}^{3}\) of aluminum weighs 2.7 grams. If a sheet of aluminum foil is \(0.0038 \mathrm{~cm}\) thick, find the weight of the roll of aluminum foil in grams.

Each of the following simplifications contains an error made by students on a test. Find the error and correct the simplification so that the expression becomes true. a. \(\left[\left(x^{2}\right)^{3}\right]^{5}=\left[x^{5}\right]^{5}=x^{25}\) b. \(\frac{7 x^{2} y^{6}}{(x y)^{2}}=\frac{7 x^{2} y^{6}}{x^{2} y^{2}}=7 x^{4} y^{8}\) c. \(\left(\frac{4 x^{3} y^{5}}{6 x y^{4}}\right)^{3}=\left(\frac{2 x^{2} y}{3}\right)^{3}=\frac{2}{3} x^{6} y^{3}\) d. \(\left(1.1 \cdot 10^{6}\right) \cdot\left(1.1 \cdot 10^{4}\right)=1.1 \cdot 10^{6}\) e. \(\frac{4 \cdot 10^{6}}{8 \cdot 10^{3}}=0.5 \cdot 10^{3}=5.0 \cdot 10^{4}\) f. \(6 \cdot 10^{3}+7 \cdot 10^{5}=13 \cdot 10^{8}\)

(Requires the use of a calculator that can evaluate powers.) \(A\) wheelchair ramp is constructed at the end of a porch, which is 4 ft off the ground. The base of the ramp is \(48 \mathrm{ft}\) from the porch. How long is the ramp? (Hint: Use the Pythagorean theorem on the inside back cover.)
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