Simplify a variety of algebraic expressions using the laws of exponents.
Example 1
Simplify 
It is important to remember that the exponent applies only to what it is written beside. The exponent 
applies only to the 
. 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}](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-ddc939b2ab2342047a67a0ff37c948a6_l3.png "Rendered by QuickLaTeX.com")
In short, 
Example 2
Simplify 
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}](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-69f734290f35f4b1f32b3d24be99cf06_l3.png "Rendered by QuickLaTeX.com")
In short, 
Example 3
Simplify 
raised to the exponent negative 
means to divide by 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}](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-640adac8dd92de85f545ef7d0095cc94_l3.png "Rendered by QuickLaTeX.com")
In short, 
Example 4
Simplify 
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}](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-0645dfeca2acbf3b3942c7ee74b3eb69_l3.png "Rendered by QuickLaTeX.com")
Practice
Try the ten questions at the end of this mathisfun page.