Question 1

To graph a piecewise function, it is essential to identify the boundaries defined by the conditions of the function. In this case, the function is divided into three segments based on the values of \$$ x \$$: less than \$$-2\$$, between \$$-2\$$ and \$$1\$$, and greater than or equal to \$$1\$$.\n\nFirst, establish vertical lines at \$$ x = -2 \$$ and \$$ x = 1 \ to $$delineate the sections of the graph. The first piece of the function is a constant value of \$$-4\$$ for \$$ x \n\nNext, for the interval \(-2 \n\nFinally, for \( x \geq 1\$$, the function is defined as \$$ f(x) = x^2 \. $$This is a parabola that opens upwards, starting at the point \$$ (1, 1) \$$ since \$$ 1^2 = 1 \. $$The parabola will continue to rise as \$$ x \$$ increases, and a solid dot is placed at \$$ (1, 1) \ to $$indicate that this point is included in the function.\n\nIn summary, the piecewise function consists of three distinct segments: a horizontal line at \$$ y = -4 \$$ for \( x

Accepted Answer

$$16$$

Question 2

What is the value of $$1^{1000}$$?

Accepted Answer

$$1$$

Question 3

Simplify the expression $$a^5$$ 	imes $$a^3.$$

Accepted Answer

$$a^{8}$$

Question 4

Which of the following is equivalent to $$x^7$$ / $$x^2$$?

Accepted Answer

$$x^{5}$$

Question 5

If $$(-3)^5 is $$evaluated, what is the result?

Accepted Answer

$$-243$$

Question 6

Which rule is used to simplify $$(y^4)^3$?

Accepted Answer

Multiply the exponents: $$y^{12}$$

Question 7

Simplify the expression $$2^{-3}.$$

Accepted Answer

$$1/8$$

Question 8

Which of the following is equivalent to $$\frac{5$$^{-2}$$}{5$$^{-5}$$}$$?

Accepted Answer

$$5^{3}$$

Question 9

A scientist is diluting a solution and the concentration halves every time it is diluted. If the initial concentration is $$C_0$$, what is the concentration after $$n$$ dilutions?

Accepted Answer

$$C_0 / 2^n$

Question 10

Which of the following is always true for any nonzero number $$a$$?

Accepted Answer

$$a^0 = 1$

Question 11

Simplify the expression $$\frac{(3^2$)^4$$}{$$3^5$$}$.

Accepted Answer

$$3^{8-5}$$ = $$3^3$$

Question 12

Which of the following is equivalent to $$\left(\frac{a}{b}\$$right)^{-3}$$?

Accepted Answer

$$\frac{b^3$}{a^3$}$$

Question 13

A radioactive substance decays so that its mass halves every hour. If the initial mass is $$M_0$$, what is the mass after $$t$$ hours?

Accepted Answer

$$M_0 / 2^t$

Question 14

Simplify the expression $$\frac{2^4$ \cdot 2$$^{-2}$$}{2^3$}.$$

Accepted Answer

$$2^{-1}$$

Question 15

Which of the following is equivalent to $$(-5)^6$$?

Accepted Answer

Question 16

If $$x^{-4}$$ = \frac{1}{$$x^4$$}$$, what is $$\left(\frac{1}{$$x^4$$}\$$right)^{-1}$$?

Accepted Answer

$$x^4$$

Question 17

Which of the following is the correct simplification of $$\frac{a^3$b^{-2}$$}{$$a^{-1}b^4$}$$?

Accepted Answer

$$a^4b$$^{-6}$

Question 18

A population of bacteria triples every 2 hours. If the initial population is $$P_0$$, what is the population after $$6$$ hours?

Accepted Answer

$$P_0$$ 	imes $$3^3$$

Question 19

Simplify the expression $$\left(\frac{2^3$}{2^5$}\$$right)^{-2}$$.

Accepted Answer

$$2^{4}$$

Question 20

Which of the following is equivalent to $$\left(-a\$$right)^{2n+1}$$ for any integer n$?

Accepted Answer

$$-a^{2n+1}$$

Rules of Exponents: Essential Properties and Applications

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