Lesson 3 · from §110 of the 1910 text

Multiplication with Any Exponents

Multiplying with exponents

The laws hold for every rational exponent, so products are found exactly as before: add the exponents of the same base, and for a power of a power, multiply the exponents.

Example 1. x^{\frac23} \cdot x^{\frac32} = x^{\frac23 + \frac32} = x^{\frac{13}{6}}.

Example 2. \sqrt[3]{x} \cdot \sqrt[4]{x^3} = x^{\frac13} \cdot x^{\frac34} = x^{\frac{13}{12}}. Change radicals to fractional exponents first.

Example 3. e^{a-3} \cdot e^{2a+1} = e^{(a - 3) + (2a + 1)} = e^{3a-2}. Exponents with letters are added like any other expressions.

Example 4. (25a^4b^6)^{-\frac12} = 25^{-\frac12}a^{-2}b^{-3} = \dfrac{1}{5a^2b^3}. Each factor is raised to the power: 25^{-\frac12} = \frac{1}{\sqrt{25}} = \frac15.

Example 5. (x^{\frac13} + y^{\frac13})(x^{\frac13} - y^{\frac13}) = x^{\frac23} - y^{\frac23}, the product of the sum and difference of two terms.

What the checker wants. Write each answer with one power for each base in a term (x^{\frac{13}{6}}, not x^{\frac23}x^{\frac32}), no power of a power, numbers worked out (512, not 2^9), and products multiplied out. Negative exponents are allowed in these answers: x^{-4} and \frac{1}{x^4} are both fine.

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Quick check

Multiply: \sqrt[3]{x} \cdot \sqrt[4]{x^3}.

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 52 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.

Perform the indicated multiplications (1–51 are oral):

x^4 \cdot x^{-2}

x^4 \cdot x^{\frac12}

x^{\frac13} \cdot x^{\frac12}

x^{\frac23} \cdot x^{\frac32}

x^{\frac12} \cdot x^{\frac13} \cdot x^0

x^{\frac34} \cdot x^{\frac43}

7x^0 \cdot x^{\frac23} \cdot x^{\frac32}

\sqrt{x} \cdot x^{\frac13}

\sqrt[3]{x} \cdot x^{\frac14}

(7x)^0 \cdot 7x^0 \cdot x^{\frac12}xy

\sqrt[3]{x} \cdot \sqrt[4]{x^3}

\sqrt{x^{-4}} \cdot \sqrt[3]{x^{-6}}

a\sqrt{ax} \cdot a^{\frac12}\sqrt{ax^{-1}}

c^2\sqrt[3]{a^3x^{-2}} \cdot c\sqrt[3]{a^3x^{-2}}

e^x \cdot e^{-x}

e^x \cdot e^{2x}

e^{a-3} \cdot e^3

e^{a-3} \cdot e^{2a+1}

e^{3-2a} \cdot e^{2+3a}

e^0 \cdot e^{3a-1} \cdot e^{3-2a}

x \cdot x^a \cdot x^b \cdot x^0 \cdot x^{2b-3a}

x^{a-b} \cdot y^a \cdot \dfrac{x^b}{y^{a-2}}

(2^3)^3

(2^3)^{-2}

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

\left[\left(\tfrac32\right)^{-3}\right]^2

(x^2)^3

\left(x^{\frac12}\right)^4

\left(x^{\frac13}\right)^{\frac12}

(x^2)^{-2}

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

\left(x^{-\frac12}\right)^{-\frac13}

(3x^{-2})^3

(5a^2)^{-3}

(c^{-2}d)^2

\left(5^0 \cdot 2^6 \cdot 3^3\right)^{\frac13}

(25a^4b^6)^{-\frac12}

\left[(2x^5)^0 \cdot 8 \cdot 4^{\frac12}\right]^{\frac12}

(a^2b)^3(a + b)

(a^2)^{3x} \cdot a^{3x}

(a^3)^{2x} \cdot (a^2)^{3x}

(a^{x+1})^2 \cdot (a^{1-x})^2

(a^3)^{x+y} \cdot (a^2)^{y-x}

(a^2b)^x \cdot (b^2a^3)^{2x}

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

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

(x^4 - a^4)x^{-2}a^{-2}

(x^2 - 5ax + 6a^2)a^{\frac23}x^{-\frac23}

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

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

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

(a^{-2} + 3)(a^{-2} - 5)