Algebra | Practice Problems, Examples & Solutions

Counting Theory

A Graphical Approach to College Algebra

Prove each statement for positive integers $n$ and $r,$ with $r \leq n$ (Hint: Use the definitions of permutations and combinations.)
$$C(n, n)=1$$

Further Topics in Algebra

Counting Theory

01:52

A Graphical Approach to College Algebra

Use any or all of the methods described in this section to solve each problem.
Committee Choices $\quad$ From 10 names on a ballot, 4 will be elected to a political party committee. In how many ways can the committee of 4 be formed if each person will have a different responsibility?

Further Topics in Algebra

Counting Theory

03:20

A Graphical Approach to College Algebra

\text {Solve each problem involving combinations.}
Delegation Choices Seven workers decide to send a delegation of 2 to their supervisor to discuss their grievances.
(a) How many different delegations are possible?
(b) If it is decided that a certain employee must be in the delegation, how many different delegations are possible?
(c) If there are 2 women and 5 men in the group, how many delegations would include at least 1 woman?

Further Topics in Algebra

Counting Theory

Mathematical Induction

4 Practice Problems

02:37

A Graphical Approach to College Algebra

- Area Show that the area of the $n$ th figure in Exercise 34 is $$\sqrt{3}\left[\frac{2}{5}-\frac{3}{20}\left(\frac{4}{9}\right)^{n-1}\right]$$

Further Topics in Algebra

Mathematical Induction

01:11

A Graphical Approach to College Algebra

Prove each statement by mathematical induction.
$\left(a^{m}\right)^{n}=a^{m n}$ (Assume that $a$ and $m$ are constant.)

Further Topics in Algebra

Mathematical Induction

02:13

A Graphical Approach to College Algebra

Use mathematical induction to prove each statement. Assume that $n$ is a positive integer.
$$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$

Further Topics in Algebra

Mathematical Induction

Negative and Rational Exponents

8 Practice Problems

03:20

A Graphical Approach to College Algebra

Factor, using the given common factor. Assume that all variables represent positive real numbers.
$$(3 r+1)^{-2 / 3}+(3 r+1)^{1 / 3}+(3 r+1)^{4 / 3} ; \quad(3 r+1)^{-2 / 3}$$

Reference: Basic Algebraic Concepts

Review of Negative and Rational Exponents

01:18

A Graphical Approach to College Algebra

Factor, using the given common factor. Assume that all variables represent positive real numbers.
$$9 z^{-1 / 2}+2 z^{1 / 2} ; \quad z^{-1 / 2}$$

Reference: Basic Algebraic Concepts

Review of Negative and Rational Exponents

01:47

A Graphical Approach to College Algebra

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers.
$$\frac{8 y^{2 / 3} y^{-1}}{2^{-1} y^{3 / 4} y^{-1 / 6}}$$

Reference: Basic Algebraic Concepts

Review of Negative and Rational Exponents