Simplify Expressions with Exponents

Simplify a variety of algebraic expressions using the laws of exponents.

laws of exponents

Example 1

Simplify 3x^4\cdot 5 x^2

It is important to remember that the exponent applies only to what it is written beside. The exponent 4 applies only to the x. Written out in full we have:

    \begin{align*}3x^4\cdot 5 x^2&=3\cdot x\cdot x\cdot x \cdot x \cdot 5 \cdot x \cdot x\\[10 pt]&=3\cdot 5\cdot x\cdot x \cdot x \cdot x \cdot x \cdot x\\[10 pt]&=15x^6\end{align}

In short, 3x^4\cdot 5 x^2=15x^6

Example 2

Simplify 5(x^4y)^3

The exponent 3 is applied to each factor in the bracket:

    \begin{align*}5(x^4y)^3&=5(x^4y)(x^4y)(x^4y)\\[10 pt]&=5\cdot x^4\cdot x^4\cdot x^4\cdot y\cdot y\cdot y\\[10 pt]&=5x^{12}y^3\end{align}

In short, 5(x^4y)^3=5x^{12}y^3

Example 3

Simplify 4^{-3}

4 raised to the exponent negative 3 means to divide by 4 three times. We ‘take the reciprocal‘. The result here is a positive number.

    \begin{align*}4^{-3}&=\\[10 pt]&=\dfrac{1}{4^3}\\[10 pt]&=\dfrac{1}{64}\end{align}

In short, 4^{-3}=\dfrac{1}{64}

Example 4

Simplify \dfrac{1}{4^{-3}}

Let’s see what the result is of a negative exponent on the denominator in general:

Simplify \dfrac{1}{a^{-m}}

    \begin{align*}\dfrac{1}{a^{-m}}&=\dfrac{1}{\frac{1}{a^m}}\\[10 pt]&=1\div\dfrac{1}{a^m}\\[10 pt]&=1 \times \dfrac{a^m}{1}\\[10 pt]&=a^m\end{align}

In short, \dfrac{1}{a^{-m}}=a^m

Applying this new law, we have a=4 and m=3.


Example 5

Simplify \dfrac{5x^7y^4}{15x^{-3}y^6}

It is helpful to gather like factors as follows:

    \begin{align*}\dfrac{5x^7y^4}{15x^{-3}y^6}&=\left(\dfrac{5}{15}\right)\left(\dfrac{x^7}{x^{-3}}\right)\left(\dfrac{y^4}{y^6}\right)\\[10 pt]&=\left(\dfrac{1}{3}\right)\left(x^{10}\right)\left(y^{-2}\right)\\[10 pt]&=\dfrac{x^{10}}{3y^2}\end{align}

Example 6

Simplify \left(\dfrac{5x^7y^4}{15x^{-3}y^6}\right)^{-2}

Let’s make a law for a fraction raised to a negative exponent:

Simplify \left(\dfrac{a}{b}\right)^{-m}

    \begin{align*}\left(\dfrac{a}{b}\right)^{-m}&=\dfrac{a^{-m}}{b^{-m}}\\[10 pt]&=a^{-m}\cdot \dfrac{1}{b^{-m}}\\[10 pt]&=\dfrac{1}{a^m}\cdot b^m\\[10 pt]&=\left(\dfrac{b}{a}\right)^m\end{align}

In short, \left(\dfrac{a}{b}\right)^{-m}=\left(\dfrac{b}{a}\right)^m

Apply this new law to example 6 we have

    \begin{align*}\left(\dfrac{5x^7y^4}{15x^{-3}y^6}\right)^{-2}&=\left(\dfrac{15x^{-3}y^6}{5x^{7}y^4}\right)^{2}\\[10 pt]&=\left(\left(\dfrac{15}{5}\right)\left(\dfrac{x^{-3}}{x^7}\right)\left(\dfrac{y^6}{y^4}\right)\right)^2\\[10 pt]&=\left(3x^{-10}y^2\right)^2\\[10 pt]&=\dfrac{9y^4}{x^{20}}\end{align}


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