QMS110 Class 2: Basic Arithmetic Part 2 – Exponent, Radicals, Decimals, Ratios, Proportions

A set of practice Q&A flashcards covering exponent basics, radical operations, decimals, fractions, ratios, proportions, and business-focused applications from the notes.

In the expression a^n, what are the base and the exponent, and how is the expression read?

The base is a, the exponent is n, and the expression a^n is read as 'a to the n' or 'a raised to the n'.

What is the exponent law for multiplying like bases: a^m * a^n = ?

a^{m+n}.

What is the exponent law for dividing like bases: a^m / a^n = ?

a^{m-n}.

How does the power change when raising a power to another power: (a^m)^n = ?

a^{mn}.

What is a^0 (assuming a ≠ 0)?

1.

What does the radical symbol √ denote, and how is a negative root written?

√ denotes the principal (positive) root; a negative root is written as -√.

In a radical like ∛x, what are the radicand and the index?

The radicand is the number under the radical sign (x); the index/order is 3 for the cube root ∛.

Can even roots of negative numbers be real numbers?

No. Even roots of negative numbers are not real; only positive real numbers have real even roots.

What does a^{1/n} represent?

The nth root of a (the number whose nth power is a).

What is the product rule for radicals: √a · √b = ?

√(ab) (for nonnegative a and b in typical real-number context).

What is the meaning of the radical index, and give an example?

The index indicates the root order; e.g., the index 3 in ∛x denotes a cube root.

What is the general approach to simplify a radical by removing perfect nth-power factors?

Factor the radicand into a product including perfect nth-power factors and pull those factors out of the radical.

What does rationalizing the denominator mean?

Multiply numerator and denominator by a radical (or conjugate) to remove radicals from the denominator.

What is the conjugate of a binomial a + b, and why is it used?

The conjugate is a − b; it is used to rationalize expressions or to simplify division by binomials with radicals.

How do you add similar radicals like terms?

Add their coefficients; only radicals with the same radicand are combinable (e.g., 3√5 + 2√5 = 5√5).

What is the product rule for radicals: (√a)(√b) = ?

√(ab) (assuming a, b ≥ 0).

What does a fractional exponent m/n mean, and how is a^{m/n} interpreted?

a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m; it represents the nth root raised to the mth power.

What is the decimal place value for 0.1 and 0.01?

0.1 is one-tenth; 0.01 is one-hundredth.

How do you rounding a decimal to a given place value?

Look at the digit to the right of the rounding place; if it is ≥5, add 1 to the rounding digit; otherwise leave it unchanged, then drop digits to the right of the rounding place.

Round 18.379 to the nearest hundredth, tenth, and integer.

Hundredth: 18.38; Tenth: 18.4; Integer: 18.

How do you convert a decimal like 2.625 to a fraction?

2.625 = 21/8 after converting 2.625 = 2625/1000 and reducing by gcd.

What is 0. repeating (0.2222…) as a fraction?

2/9.

What is a ratio?

A fixed relationship between two or more quantities; expressed in formats such as a:b, a/b, a to b, or as a percentage.

Name four formats to express a ratio.

Colon (2:1), fraction (2/1), decimal (2.0), and percent (200%).

What is a proportion and what key property does it have?

An equation relating two ratios; it satisfies ad = bc when a/b = c/d (cross-multiplication).

What is cross-multiplication in a proportion with a/b = c/d?

ad = bc.

What is the general method to allocate overhead costs proportionally by direct costs?

Compute total direct costs; allocate overheads in the same ratio as direct costs: Overheadi = (Directi / Sum Directs) × Total Overhead.

In the business overhead example, what are the allocated overheads Co, Wo, No (rounded to two decimals) for direct costs Chocolate 743,682; Wafers 2,413,795; Nougat 347,130 with total overhead 721,150?

Co ≈ 153,028.93; Wo ≈ 496,691.43; No ≈ 71,429.63; Totals: Ct ≈ 896,710.93; Wt ≈ 2,910,486.43; Nt ≈ 418,559.64.

What is the purpose of the conjugate in rationalizing a denominator?

To create a difference of squares in the denominator, removing radicals when multiplied by the conjugate.

What is an example of adding and simplifying radicals with literal numbers, as shown in the notes (e.g., 8√? or similar)?

Radicals with literal numbers can be added or combined by aligning like terms, e.g., a combination like 2√a + 3√a = 5√a.