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Chapter 12: Problem 3

Express cach of the following expressions using a single positive index: (a) \(x^{4} x^{7}\) (b) \(x^{2}(-x)\) (c) \(\frac{x^{2}}{x}\). (d) \(\frac{x^{-2}}{x^{-1}}\) (c) \(\left(x^{-2}\right)^{4}\) (f) \(\left(x^{-25} x^{-3.5}\right)^{2}\)

Short Answer

Expert verified
Question: Express the following expressions using a single positive index: a) \(x^4 x^7\) b) \(x^2(-x)\) c) \(\frac{x^2}{x}\) d) \(\frac{x^{-2}}{x^{-1}}\) e) \(\left(x^{-2}\right)^{4}\) f) \(\left(x^{-25} x^{-3.5}\right)^{2}\) Answer: a) \(x^{11}\) b) \(x\) c) \(x\) d) \(x^{-1}\) e) \(x^{-8}\) f) \(x^{-57}\)

Step by step solution

01

Problem (a) Solution

We are given expression \(x^4 x^7\). Using the first property of exponents, we add the exponents: \(x^4 \cdot x^7 = x^{4 + 7} = x^{11}\).
02

Problem (b) Solution

We are given expression \(x^2(-x)\). We can rewrite \((-x)\) as \(x^{-1}\). Now, we can apply the first property of exponents: \(x^2 \cdot x^{-1} = x^{2 + (-1)} = x^{1} = x\).
03

Problem (c) Solution

We are given expression \(\frac{x^2}{x}\). Using the second property of exponents, we subtract the exponents: \(\frac{x^2}{x} = x^{2 - 1} = x^1=x\).
04

Problem (d) Solution

We are given expression \(\frac{x^{-2}}{x^{-1}}\). Using the second property of exponents, we have: \(\frac{x^{-2}}{x^{-1}} = x^{-2 - (-1)}= x^{-2+1} = x^{-1}\).
05

Problem (e) Solution

We are given expression \(\left(x^{-2}\right)^{4}\) . Using the third property of exponents, we have: \(\left(x^{-2}\right)^{4} = x^{-2 \cdot 4}=x^{-8}\).
06

Problem (f) Solution

We are given expression \(\left(x^{-25} x^{-3.5}\right)^{2}\). First, we simplify the expression inside the parentheses using the first property of exponents: \(x^{-25} x^{-3.5} = x^{-25 - 3.5} = x^{-28.5}\). Next, we apply the third property of exponents to find the final answer: \(\left(x^{-28.5}\right)^{2} = x^{-28.5 \cdot 2} = x^{-57}.\) We have now written all of the given expressions using a single positive index.

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Most popular questions from this chapter

Evaluate (a) \(2^{3}\) (b) \(3^{2}\) (c) \(\frac{5^{13}}{5^{12}}\) (d) \(\frac{19^{-11}}{19^{-13}}\) (c) \(\left(2^{1 / 4}\right)^{8}\) (f) \((-4)^{-2}\) (g) \(4^{-1 / 2}\) (h) \(\left(9^{1 / 3}\right)^{3 / 2}\) (i) \(\sqrt{32} \sqrt{2}\) (j) \(\sqrt{0.01}\) (k) \(81^{3 / 4}\)

The fracture strength of glass may be increased by etching away a thin surface layer. It is believed that the etching may alter surface crack geometry (i.e., reduce the crack length and increase the tip radius). Compute the ratio of the etched and origi- nal crack-tip radii for a fourfold increase in frac- ture strength if half of the crack length is removed.

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A hypothetical AX type of ceramic material is known to have a density of 2.10 g/cm3 and a unit cell of cubic symmetry with a cell edge length of 0.57 nm. The atomic weights of the A and X elements are 28.5 and 30.0 g/mol, respectively. On the basis of this information, which of the following crystal structures is (are) possible for this material: sodium chloride, cesium chloride, or zinc blende? Justify your choice(s).

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