To write three expressions equivalent to

x^{- 2},

we can think of the following power rules.

x^{m + n} x^{m \times n} x^{m - n} = x^{m} x^{n} = (x^{m})^{n} = \frac{x ^{m}}{x ^{n}} (multiplication) (powers) (division)

By rewriting the expression in different ways that match the exponents on the left-hand side, we can write three different single-term expressions that are equivalent to

x^{- 2} .

Multiplication

By rewriting the exponent as a sum we can write a new equivalent expression.

x^{- 2}

WriteSum

Write as a sum

x^{2 + (- 4)}

SumInExponent

$a^{m + n} = a^{m} \cdot a^{n}$

x^{2} \cdot x^{- 4}

Powers

By rewriting the exponent as a product, we can write a new equivalent expression.

x^{- 2}

SplitIntoFactors

Split into factors

x^{0.5 (- 4)}

ProdInExponent

$a^{m \cdot n} = (a^{m})^{n}$

(x^{0.5})^{- 4}

$a^{0.5} = a$

(x)^{- 4}

Division

By rewriting the exponent as a difference we can write a new equivalent expression.

x^{- 2}

WriteDiff

Write as a difference

x^{2 - 4}

DiffInExponent

$a^{m - n} = \frac{a ^{m}}{a ^{n}}$

\frac{x ^{2}}{x ^{4}}

Exercise 97 Page 98

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Multiplication

Powers

Division