Q: A student claims that $$a^0 = 0$ for any a$. Which of the following best explains why this is incorrect?

Because $$a^0 = 1$ for any a$$ \neq 0$$

Question 1

Which of the following correctly simplifies $$5^{-2}$$ and expresses the answer with positive exponents?

Accepted Answer

$$\frac{1}{25}$$

Question 2

A biologist is studying bacteria growth. The population doubles every hour, so after $$n$$ hours, the population is $$P = 2^n. $$What is the value of $$2^0$$ and what does it represent in this context?

Accepted Answer

$$1$$; the initial population before any growth

Question 3

Simplify the expression $$\frac{6^7$}{6^3$}$$ using properties of exponents.

Accepted Answer

$$6^4$$

Question 4

Which of the following is equivalent to $$(x^{-2}$$)^{-5}$, written with positive exponents?

Accepted Answer

$$x^{10}$$

Question 5

A chemist dilutes a solution so that its concentration is reduced by a factor of $$10$$ for each step. If the initial concentration is $$C_0$$, what is the concentration after $$n$$ steps, and what does $$C_0$$ \cdot $$10^{-2}$$ represent?

Accepted Answer

After $$n$$ steps: $$C_0$$ \cdot $$10^{-n}$$; $$C_0$$ \cdot $$10^{-2} is $$the concentration after 2 steps

Question 6

Simplify the expression $$(3x^5)(5x^2$)$.

Accepted Answer

$$15x^7$$

Question 7

Which of the following is the correct simplification of $$\frac{a$$^{10}$$b^6$$}{$$a^8b$$}$?

Accepted Answer

$$a^2b^5$

Question 8

A medical researcher is analyzing the decay of a radioactive substance, which halves every hour. If the initial amount is $$A_0$$, what is the amount after $$n$$ hours, and how can this be written using exponents?

Accepted Answer

$$A_0$$ \cdot (\frac{1}{2})^n$$

Question 9

Simplify $$(2x^3y^5$)^4$$ and write your answer with positive exponents.

Accepted Answer

$$16x^{12}y$$^{20}$

Question 10

Which of the following is equivalent to $$\frac{7a^2$}{d$$^{-10}$$}$$ with only positive exponents?

Accepted Answer

$$7a^2d$$^{10}$

Question 11

Simplify $$(\frac{6x$$^{-5}$$}{4y^3$})^2$ and write your answer with positive exponents.

Accepted Answer

$$\frac{36}{16}x$$^{-10}$$y^{-6}$$ = \frac{9}{$$4x^{10}y^6$}$$

Question 12

Which of the following is the correct simplification of $$(x^2)(x^8$)$?

Accepted Answer

$$x^{10}$$

Question 13

Simplify $$(3x^4y$$^{-2}$$z^6)(3x$$^{-2}$$y^{-5}z$$)$$ and write your answer with positive exponents.

Accepted Answer

$$9x^2z^7$y^{-7}$$, which is $$\frac{9x^2$z^7$$}{$$y^7$$}$

Question 14

Which of the following is undefined?

Accepted Answer

$$0^0$$

Question 15

Simplify $$(-3x^5y$$^{-1}$$)^{-4}$$ and write your answer with positive exponents.

Accepted Answer

$$\frac{1}{81x$$^{20}$$y^{-4}$$} = \frac{$$y^4$$}{$$81x^{20}$$}$

Question 16

A physics formula involves $$a^{-n}$$, where $$a 
eq 0. $$Which of the following is equivalent to $$a^{-n}$$?

Accepted Answer

$$\frac{1}{a^n}$$

Question 17

Simplify $$\left(\frac{x^2$}{2y}\$$right)^4$$ and write your answer with positive exponents.

Accepted Answer

$$\frac{x^8$}{16y^4$}$$

Question 18

Which of the following is the correct simplification of $$\frac{x^3$}{x^7$}$$?

Accepted Answer

$$x^{-4}$$

Question 19

Simplify $$(x^{-1})^5$ and write your answer with positive exponents.

Accepted Answer

$$\frac{1}{x^5$}$$

Question 20

A student claims that $$a^0 = 0$ for any a$. Which of the following best explains why this is incorrect?

Accepted Answer

Because $$a^0 = 1$ for any a$$ 
eq 0$$

Section 1.6: Properties of Exponents – Intermediate Algebra Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards