Positive Exponents and Expressions Without Denominators

Letters with zero and negative exponents

In this chapter the letters in a fraction may not take on such values as would make any denominator zero, or any expression zero to the zero power.

Example 1. Write \dfrac{3a^{-2}bc^0}{4x^{-3}y} with positive exponents.

c^0 = 1; the factor a^{-2} moves to the denominator as a^2, and x^{-3} moves to the numerator as x^3:

\frac{3a^{-2}bc^0}{4x^{-3}y} = \frac{3bx^3}{4a^2y}.

Example 2. Simplify \dfrac{2}{a^{-2} - b^{-2}}.

Here a^{-2} is a term, not a factor, of the denominator, so it cannot simply be moved. Write the negative powers as fractions first:

\frac{2}{a^{-2} - b^{-2}} = \frac{2}{\frac{1}{a^2} - \frac{1}{b^2}} = \frac{2}{\frac{b^2 - a^2}{a^2b^2}} = \frac{2a^2b^2}{b^2 - a^2}.

Writing without a denominator. By the same rule, every factor of a denominator can be moved up into the numerator with its exponent's sign changed:

\frac{3x^2}{a^2y^3} = 3a^{-2}x^2y^{-3}, \qquad \frac{5}{c(x - y)^2} = 5c^{-1}(x - y)^{-2}.

A plain number in the denominator may stay as a fraction coefficient, \frac34x^2, or be written 3 \cdot 4^{-1}x^2.

Typing answers. For "write with positive exponents", type fractions with /: 3b x^3/(4a^2 y). The checker marks an answer that still has a negative or zero exponent as "equal, but write it with positive exponents only". For "write without a denominator", type negative exponents: 3a^(-2)x^2y^(-3), 5c^(-1)(x - y)^(-2).

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Quick check

Write \dfrac{3a^{-2}bc^0}{4x^{-3}y} with positive exponents.

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 40 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.

Write with positive exponents and simplify results:

m^{-3}

2a^{-3}

3ab^{-2}

7x^2y^{-2}

x^{-1}y^{-2}z

4a^3b^{-2}c^2

\dfrac{3}{a^{-2}}

\dfrac{4x}{y^{-3}}

\dfrac{4c^0}{xy^{-2}}

\dfrac{4a^{-2}b^0}{y^{-5}}

\dfrac{5^{-1}(ab)^0}{10^{-2}b^2}

\dfrac{3a^3b^{-2}c}{4a^{-2}b^0}

\dfrac{12x^2y^{-1}}{2yx^{-1}}

\dfrac{10^{-1}a}{bc^3}

\dfrac{4a^{-2}bc^{-3}}{6a^{-2}b^{-3}c^0}

\dfrac{4^{-3}r^{-6}s^6}{s^{-2}r^{-2}t^3}

\dfrac{2s^{-1}}{m^na^{-b}}

\dfrac{5x^2y^a}{x^{-m}y^{-4}}

\dfrac{2}{a^{-2} - b^{-2}}

\dfrac{3}{a^{-1} + b^{-1}}

\dfrac{a}{a^{-2} - b^{-2}}

\dfrac{5se^2}{s^{-2} + e^{-2}}

\dfrac{a^{-2}}{a^{-2} + b^{-2}}

\dfrac{a^{-3}b^{-3}}{a^{-3} + b^{-3}}

\dfrac{a^{-2} - b^{-2}}{a^{-1} + b^{-1}}

\dfrac{a^{-4} - b^{-4}}{a^{-2} - b^{-2}}

\dfrac{a^{-1} + b^{-1}}{a^{-3} + b^{-3}}

\dfrac{a^{-3} - 27^{-1}}{a^{-1} - 3^{-1}}

Write without a denominator:

\dfrac{2xy}{z^2}

\dfrac{4a^3}{b^3}

\dfrac{3x}{a^{-2}b^4}

\dfrac{4sc^{-3}}{2^{-1}s^{-2}}

\dfrac{12a^2b^3}{4xy^2}

\dfrac{7x^{-1}y^2}{2^{-1}y^3}

\dfrac{5ac^{-2}}{c(x - y)^2}

\dfrac{7m^{-3}n^{\frac12}}{m^{\frac23}(m - n)^0}

\dfrac{1}{5a^2(c + d)^{-3}}

\dfrac{a(x - y)^{-2}}{bcx(x - y)}

\dfrac{42m^{-n}n^{2m}}{56m^{-2n}n^{-3n}}

\dfrac{r^{-1}s^2}{r^{-2}s^3(s - r)^3}