Exponents: learn and practice

An exponent counts repeated factors

An exponent tells how many times to use a number as a factor. In 2 to the third power, the base is 2 and the exponent is 3, so you multiply 2 × 2 × 2. That product is 8. It is not 2 × 3, and it is not 2 + 2 + 2. The exponent counts factors, not addends. x squared means x × x. x to the first power is just x, one factor. A linear equation keeps the unknown to the first power. This page starts the next step, when a product of two sums produces an x squared term.

Every term in one sum meets every term in the other

A binomial is a sum of two terms, such as x + 2. The product of two binomials is not found by multiplying only the first terms and only the second terms. Each term in the first binomial multiplies each term in the second. (x + 2)(x + 3) has four products: x times x, x times 3, 2 times x, and 2 times 3. Then you add those four products and combine the terms that share the same power of x.

Worked example 1: a whole-number power

Evaluate 2 to the fourth power. The base is 2 and the exponent is 4, so there are four factors of 2. 2 × 2 = 4. 4 × 2 = 8. 8 × 2 = 16. So the value is 16. Multiplying the base by the exponent gives 2 × 4 = 8, which is a different operation and the wrong value. The check is the count of factors: four 2s in a product, not four 2s in a sum.

Worked example 2: the square of a sum

Expand (x + 1) squared, which means (x + 1)(x + 1). Four products: x times x is x squared, x times 1 is x, 1 times x is x, and 1 times 1 is 1. Add them: x squared + x + x + 1. The two middle terms are like terms, so x + x = 2x. The expansion is x squared + 2x + 1. A common slip is to square each term and stop, writing x squared + 1. That drops the middle term. The middle term is there because each part of the first sum meets each part of the second.

Worked example 3: expand (x+2)(x+3)

Expand (x + 2)(x + 3). Four products: x times x is x squared, x times 3 is 3x, 2 times x is 2x, and 2 times 3 is 6. Add them: x squared + 3x + 2x + 6. Combine 3x + 2x to get 5x. The expansion is x squared + 5x + 6. Check with a number. If x is 1, (1 + 2)(1 + 3) = 3 × 4 = 12, and 1 + 5 + 6 = 12. If x is 0, both sides are 6. The middle term is the sum of the two cross products.

List the four products, then combine

Write the four products before you simplify, so a missing cross term is easy to see. Combine only like powers. Then substitute a small whole number for x in the original product and in your expansion. The two values have to match. After that, expand a new pair of binomials on the board.

Common questions about exponents

Is 2 to the third power the same as 2 times 3?

No. The exponent counts factors. 2 to the third power is 2 × 2 × 2, which is 8. 2 times 3 is 6.

Why does (x+2)(x+3) have an x squared term?

Because the x in the first binomial multiplies the x in the second. x times x is x squared. The other three products are 3x, 2x, and 6.

Can I square each part and add?

No. (x + 1) squared is not x squared + 1. The cross products add a middle term, so the expansion is x squared + 2x + 1.