Exercises 1

This page allows you to practice some exercises on Algebra. If you notice that you have difficulties, we advise you to go over the corresponding sections in the book.

These questions correspond to the following sections: - Section 2.1 - The Real Numbers - Section 2.2 - Integer Powers - Section 2.3 - Rules of Algebra

Question 1

Which of the following statements are true? (a) $350$ is a natural number.

(b) $-3$ is to the right of $-1$ on the number line.

(c) $-17$ is a natural number.

(d) There is no natural number that is not rational.

(e) $3.1415$ is not rational.

(f) The sum of two irrational numbers is irrational.

(g) $-2/5$ is rational.

(h) All rational numbers are real.

Show answer

(a) True. (b) False, $-3$ is less than $-1$, so this value is located to the left of $-1$ on the number line. (c) False, all natural numbers are positive. This number belongs to the set of integers. (d) True, every natural number is rational. (e) False, since $3.1415 = 31415/10000$, i.e. the quotient of two integers. (Note that we are not dealing with $\pi$, but solely an approximation to this irrational number). (f) False. An easy counterexample is: $\sqrt{2} + (-\sqrt{2}) = 0.$ (g) True, as it can be written as $p/q$, where $p$ and $q$ are integers, $q \neq 0$. (h) True.

Question 2

Simplify the following expressions: (a) $x^p x^{2p}$, where $p$ is an integer

(b) $t^s : t^{s-1}$, where $t \neq 0$ and $s$ is an integer

(c) $a^2 b^3 a^{-1} b^5$, where $a \neq 0$

(d) $\frac{t^p t^{q-1}}{t^r t^{s-1}}$, where $t \neq 0$ and $p, q, r, s$ are integers

Show answer

(a) $x^p x^2p = x^{p+2p} = x^{3p}$

(b) $t^s : t^{s-1} = t^{s-(s-1)} = t^{s-s+1} = t^1 = t$

(c) $a^2b^3a^{-1}b^5 = a^{2-1} b^{3+5} = ab^8$

(d) $\frac{t^p t^{q-1}}{t^r t^{s-1}} = t^{p+q-1-(r+s-1)} = t^{p+q-r-s}$

Question 3

Expand and simplify the following expressions: (a) $\frac{a^4 b^{-3}}{(a^2 b^{-3})^2}$

(b) $\frac{p^{\gamma} (pq)^{\sigma}}{p^{2\gamma + \sigma} q^{\sigma - 2}}$

(c) $\frac{(a+1)^2 - (a-1)^2}{(b+1)^2 - (b-1)^2}$

(d) $(a+b)^3$

Show answer

(a) $\frac{a^4 b^{-3}}{(a^2 b^{-3})^2} = \frac{a^4 b^{-3}}{a^4 b^{-6}} = b^{-3 -(-6)} = b^3$

(b) $\frac{p^{\gamma} (pq)^{\sigma}}{p^{2\gamma + \sigma} q^{\sigma - 2}} = p^{-\gamma}q^2$

(c) $\frac{(a+1)^2 - (a-1)^2}{(b+1)^2 - (b-1)^2} = \frac{a^2 +2a + 1 - (a^2 - 2a + 1)}{b^2 +2b + 1 - (b^2 - 2b +1)} = \frac{4a}{4b} = \frac{a}{b}$

(d) $(a+b)^3 = (a+b)^2 (a+b) = (a^2 + 2ab + b^2)(a+b) = a^3 + 3a^2 b + 3a b^2 + b^3$

Question 4

Factor the following expressions:

(a) $y^2 - 10y + 25$

(b) $x^2 - y^2$

(c) $16a^2 + 16ab + 4b^2$

(d) $z^3 y - 4(zy)^2 + 4z y^3$

Show answer

(a) $(y-5)(y-5) = (y-5)^2$ (b) $(x+y)(x-y)$ (c) $2 \cdot 2 (2a + b) (2a + b) = 4(2a+b)^2$ (d) $zy(z - 2y)(z - 2y) = zy(z - 2y)^2$