Expanding and Longer Products

Expanding with exponents

Expanding. Use the familiar forms (t \pm u)^2 = t^2 \pm 2tu + u^2 and (t + u)^3 = t^3 + 3t^2u + 3tu^2 + u^3, with the laws of exponents for each term.

Example 1. (a^{-1} - a)^2 = a^{-2} - 2a^{-1}a + a^2 = a^{-2} - 2 + a^2.

Longer products are multiplied term by term, exactly as polynomials are, adding exponents of each letter. Many of them are special products in disguise:

\left(a^{\frac45} - 2a^{\frac25}x + 4x^2\right)\left(a^{\frac25} + 2x\right) = a^{\frac65} + 8x^3,

the sum of two cubes, with t = a^{\frac25} and u = 2x.

What the checker wants: one power for each base in a term, numbers worked out, and everything multiplied out with like terms combined.

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Quick check

Expand (a^{-1} - a)^2.

How to type exponents

A power: x^3. A negative or fractional exponent goes in parentheses: x^(-2), x^(2/3), a^(-1/2). (You may write x^-2, but x^-1/2 means \frac{x^{-1}}{2}, so use x^(-1/2).)
An exponent with letters: e^(2x), x^(a+1), m^(n-1).
Radicals are accepted too: sqrt(x), cbrt(x), root(x, 4).

Exercises

0 of 19 solved

How to type math

Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.

Expand:

(a^{-1} - a)^2

(a^3 - 2a^{-2})^3

(a^{-1} - 2a + 3a^{-2})^2

(e^x + e^{-x})^2

(a^{-1} + b^{-2})\left(\dfrac1a - \dfrac{1}{b^2}\right)

(e^{2x} - 2 + e^{-2x})^2

\left(3a^{-\frac12} + 2a^{\frac23}\right)^2

Perform the indicated multiplications:

\left(a^{\frac45} - 2a^{\frac25}x + 4x^2\right)\left(a^{\frac25} + 2x\right)

\left(x^{\frac13} + 2y^{\frac13}\right)\left(x^{\frac23} - 2x^{\frac13}y^{\frac13} + 4y^{\frac23}\right)

\left(a - a^{\frac12}b^{\frac12} + b\right)\left(a + a^{\frac12}b^{\frac12} + b\right)

\left(a^{\frac12} + a^{\frac14}b^{\frac14} + b^{\frac12}\right)\left(a^{\frac12} - a^{\frac14}b^{\frac14} + b^{\frac12}\right)

\left(x - x^{\frac12}y^{-\frac12} + y^{-1}\right)\left(x + x^{\frac12}y^{-\frac12} + y^{-1}\right)

\left(\sqrt[3]{a^5} - 3\sqrt{a}\right)\left(\sqrt[3]{a^5} - 3\sqrt{a}\right)

\left(5\sqrt{c^{-5}}\,\sqrt[3]{d^{-3}} - a\sqrt[5]{c^{-2}}\right)^3

\left(\sqrt a + \dfrac{\sqrt b}{c}\right)\left(\sqrt a + \dfrac{\sqrt b}{c}\right)^2

\left(m^{n-1} - 5m^{n-2}a^n + 25a^{2n}\right)(m + 5a^n)

\left(r^{\frac52} - r^2 + r^{\frac32} - r + r^{\frac12} - 1\right)\left(r^{\frac12} + 1\right)

\left(16a^{\frac45} + 8a^{\frac35}b^{\frac12} + 4a^{\frac25}b + 2a^{\frac15}b^{\frac32} + b^2\right)\left(2a^{\frac15} - b^{\frac12}\right)

\left(25x^{-4} + 15x^{-2}y^{-8} + 9y^{-16}\right)\left(5x^{-2} - 3y^{-8}\right)