Algebraic Expressions, Powers and Roots

Algebra is arithmetic with letters standing in for some of the numbers. Its rules are the rules of arithmetic, written so that they hold for every number at once.

What does a letter stand for?

A letter stands for a number that is not fixed, or not known yet. 3n means 3 × n, $n^2$n² means n × n, and $\frac{n}{4}$n/4 means $n/4$n / 4. See Algebraic Notation.

Substitution puts a number where the letter stands: if n = 4, then 3n + 2 = 14. See Substitution.

To solve an equation, do the same thing to both sides until the letter stands alone. For 3x + 7 = 22, subtract 7 from both sides, then divide both sides by 3: x = 5. See Solving Linear Equations and the equations and inequalities guide.

How do you collect like terms?

Like terms have exactly the same letter part. Add the numbers in front and keep the letter part: 3x + 5x = 8x. Terms with different letter parts stay separate, so 3x + 5y and $3x+3x^2$3x + 3x² cannot be simplified. See Like Terms.

Expanding brackets

To expand a bracket, multiply every term inside by the term outside: 4(2x + 3) = 8x + 12. This is the distributive law, the same fact that makes 4 × 23 = 80 + 12. See Expanding Brackets.

With two brackets, every term in the first multiplies every term in the second, and then the like terms collect: $\left(x+3\right)\left(x+5\right)=x^2+5x+3x+15=x^2+8x+15$(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15. See Expanding Two Brackets.

How do you factor?

Factoring writes a sum as a product; it is expanding in reverse. Look for a common factor first: 6x + 9 = 3(2x + 3). See Factoring by Common Factor.

To factor $x^2+bx+c$x² + bx + c, find two numbers that multiply to c and add to b. For $x^2+8x+15$x² + 8x + 15, 3 × 5 = 15 and 3 + 5 = 8, so it is (x + 3)(x + 5). See Factoring Quadratics.

With a number in front of $x^2$x², the pair must multiply to a × c. For $2x^2+11x+15$2x² + 11x + 15 that is 30, and 5 + 6 = 11. Split the middle term as 6x + 5x and factor each pair: 2x(x + 3) + 5(x + 3). The bracket (x + 3) appears twice, so take it out: (2x + 5)(x + 3). See Factoring by Grouping and Factoring Quadratics with a Leading Coefficient.

The identities worth recognizing

Three expansions are worth knowing by sight. See Algebraic Identities.

• ${\left(a+b\right)}^2=a^2+2ab+b^2$(a + b)² = a² + 2ab + b²
• ${\left(a-b\right)}^2=a^2-2ab+b^2$(a − b)² = a² − 2ab + b²
• $a^2-b^2=\left(a+b\right)\left(a-b\right)$a² − b² = (a + b)(a − b), the difference of two squares, which makes $97\times 103=100^2-3^2=9$97 × 103 = 100² − 3² = 9,991.

A sum or difference of two cubes factors too: $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$a³ + b³ = (a + b)(a² − ab + b²). See The Sum and Difference of Two Cubes. If two expressions are equal for every x, their $x^2$x² terms, x terms and constant terms are each equal, which gives one equation per unknown. See Finding Unknown Coefficients by Matching.

What are the laws of indices?

An index counts how many copies of the base are multiplied together: $2^3=2\times 2\times 2=8$2³ = 2 × 2 × 2 = 8, not 6. Multiplying powers of the same base adds the indices: $x^3\times x^4=x^7$x³ × x⁴ = x⁷, three copies beside four. Dividing subtracts them: $x^5/x^2=x^3$x⁵ / x² = x³. The bases must match, so $2^3\times 3^2$2³ × 3² is just 8 × 9. See Index Notation and Laws of Indices.

A power of a power multiplies the indices: ${\left(x^3\right)}^4=x^{12}$(x³)⁴ = x¹², four groups of three copies. See The Power of a Power Rule.

The subtraction rule gives $2^3/2^3=2^0$2³ / 2³ = 2⁰, and division gives 8 ÷ 8 = 1, so $2^0=1$2⁰ = 1. One step further, $2^2/2^3=2^{-1}=\frac{1}{2}$2² / 2³ = 2⁻¹ = 1/2: a negative index means one over the power. See Zero and Negative Indices.

A square root undoes squaring, so $\sqrt{25}=5$√25 = 5; a cube root undoes cubing, so $\sqrt[3]{27}=3$∛27 = 3. See Squares and Square Roots and Cubes and Cube Roots.

A fractional index is a root: $x^{\frac{1}{2}}\times x^{\frac{1}{2}}=x^1$x^(1/2) × x^(1/2) = x¹, so $x^{\frac{1}{2}}$x^(1/2) is the square root of x, and $8^{\frac{2}{3}}={\left(\sqrt[3]{8}\right)}^2=4$8^(2/3) = (∛8)² = 4. See Fractional Indices.

A power is worked out after brackets and before multiplying, so $3\times 2^2=12$3 × 2² = 12, not 36. See Powers in the Order of Operations.

Standard form

Standard form writes a number as a value from 1 up to 10 times a power of ten: 4,700,$000=4.7\times 10^6$000 = 4.7 × 10⁶ and $0.00032=3.2\times 10^{-4}$0.00032 = 3.2 × 10⁻⁴. See Standard Form.

To multiply, multiply the front numbers and add the powers, then bring the front number back below 10: $\left(3\times 10^5\right)\times \left(4\times 10^3\right)=12\times 10^8=1.2\times 10^9$(3 × 10⁵) × (4 × 10³) = 12 × 10⁸ = 1.2 × 10⁹. See Calculating in Standard Form.

What is a surd?

A surd, also called a radical, is a root that is not a whole number or a fraction. Its decimal never stops, so $\sqrt{2}$√2 is exact and 1.414 is only an approximation. To simplify a surd, split the number into a square factor and what is left: $\sqrt{12}=\sqrt{4}\times \sqrt{3}=2\sqrt{3}$√12 = √4 × √3 = 2√3. See Radicals.

Surds with the same root add like terms: $2\sqrt{3}+5\sqrt{3}=7\sqrt{3}$2√3 + 5√3 = 7√3, while $\sqrt{2}+\sqrt{3}$√2 + √3 stays as it is. Multiplying joins the roots: $\sqrt{2}\times \sqrt{8}=\sqrt{16}=4$√2 × √8 = √16 = 4. See Adding and Multiplying Surds.

To rationalize a denominator, multiply the numerator and the denominator by the root: $\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$1/√3 = √3/3. If the denominator is $2+\sqrt{5}$2 + √5, multiply both by $2-\sqrt{5}$2 − √5 instead; the difference of two squares makes the denominator 4 − 5 = −1, so $\frac{1}{\left(2+\sqrt{5}\right)}=\sqrt{5}-2$1/(2 + √5) = √5 − 2. See Rationalizing the Denominator.

Algebraic fractions

An algebraic fraction follows the ordinary fraction rules, with one extra step: factor before you cancel, because only a factor of the whole numerator and denominator can be canceled: $\frac{\left(x^2-9\right)}{\left(x+3\right)}=\frac{\left(x+3\right)\left(x-3\right)}{\left(x+3\right)}=x-3$(x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3. See Algebraic Fractions.

Adding needs a common denominator, just as $\frac{1}{2}+\frac{1}{3}$1/2 + 1/3 does: $\frac{1}{x}+\frac{1}{y}=\frac{\left(x+y\right)}{xy}$1/x + 1/y = (x + y)/xy. See Adding and Multiplying Algebraic Fractions.

One expression, simplified step by step

Simplify ${\left(2x^2{y}^3\right)}^3/\left(4x^4{y}^2\right)$(2x²y³)³ / (4x⁴y²). Work the bracket first, then the numbers, then each letter.

1. Expand the power. The index outside applies to every factor inside, including the 2: ${\left(2x^2{y}^3\right)}^3=8x^6{y}^9$(2x²y³)³ = 8x⁶y⁹.
2. Divide the numbers: 8 ÷ 4 = 2.
3. Subtract the indices, letter by letter: $x^6/x^4=x^2$x⁶ / x⁴ = x² and $y^9/y^2=y^7$y⁹ / y² = y⁷.
4. Write the answer: $2x^2{y}^7$2x²y⁷.

Check by substitution. With x = 1 and y = 2, the original gives $16^3/16=256$16³ / 16 = 256 and the answer gives 2 × 128 = 256. One agreement does not prove the answer, but a disagreement proves a mistake.

The four mistakes worth naming

• Losing the middle term. ${\left(a+b\right)}^2$(a + b)² is $a^2+2ab+b^2$a² + 2ab + b², not $a^2+b^2$a² + b².

• Canceling a term. In $\frac{\left(x+3\right)}{3}$(x + 3)/3 the 3s do not cancel; 3 is not a factor of the whole numerator.
• Adding indices with different bases. $x^3\times y^3$x³ × y³ is ${\left(xy\right)}^3$(xy)³, not $x^6$x⁶.

• Leaving standard form out of range. $12\times 10^8$12 × 10⁸ tidies to $1.2\times 10^9$1.2 × 10⁹.

Where this leads next

These rules are the tools for solving equations, the subject of equations and inequalities. Whether a root is irrational is settled in number theory, and the arithmetic these rules generalize is in arithmetic methods.

Practice it in the game

Math Challenge teaches each of these lessons with its drawings and practice questions, and a wrong answer re-teaches the step it came from.

The Powers topic drills the index laws.