In a term like $7x^2$ , two numbers do different jobs: the coefficient says how many, and the exponent says how many times the base is multiplied. Every rule below moves one of them and leaves the other alone.

Concept Two parts of a term

The coefficient is a count sitting in front of the term. The exponent is a small raised number that only records repeated multiplication of the base. Keep them separate and the rules stop clashing.

Concept Adding is not multiplying

Repeat $x$ by addition and you build up a coefficient. Repeat it by multiplication and you build up an exponent. Same four copies of $x$ , two different answers.

x + x + x + x = 4x

x \cdot x \cdot x \cdot x = x^{4}

Concept Like terms add their coefficients

Terms are like when the variable part matches exactly, exponent included. Add them by adding the coefficients; the $x^{2}$ part is carried along unchanged.

x^{2} + x^{2} + x^{2} = 3x^{2}

7x^{2} + 13x^{2} = 20x^{2}

The exponent never moves here. $7x^{2} + 13x^{2}$ is not $20x^{4}$ .

Concept A power raised to a power

$(x^{3})^{4}$ means four copies of $x^{3}$ multiplied together, which is $x$ written twelve times. Multiply the exponents.

(x^{3})^{4} = x^{3 \times 4} = x^{12}

Concept Dividing powers with the same base

Two jobs, done separately: divide the coefficients, subtract the exponents.

\dfrac{12x^{9}}{3x^{4}} = \dfrac{12}{3}\,x^{\,9-4} = 4x^{5}

Concept The zero exponent

Anything non-zero raised to the power $0$ is $1$ . The exponent on the inside never gets a chance to matter.

(y^{6})^{0} = 1

Not $y^{6}$ , and not $0$ . Multiplying the exponents would give $y^{0}$ , and $y^{0} = 1$ .

Summary

$x + x + x + x = 4x$ , but $x \cdot x \cdot x \cdot x = x^{4}$ .
Like terms add their coefficients: $7x^{2} + 13x^{2} = 20x^{2}$ .
Power to a power multiplies the exponents: $(x^{3})^{4} = x^{12}$ .
Dividing the same base: divide coefficients, subtract exponents: $\dfrac{12x^{9}}{3x^{4}} = 4x^{5}$ .
Any non-zero base to the power $0$ is $1$ : $(y^{6})^{0} = 1$ .

Basic Algebra