Properties of Exponents - Lesson

Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.

LaTeX: x^n\cdot x^m=x^{n+m}xn⋅xm=xn+m

Examples

LaTeX: 5^2+5^7=\:5^{2+7\:}=\:5^952+57=52+7=59
LaTeX: x^{2\:}+x^3=x^{2+3}\:=\:x^5x2+x3=x2+3=x5
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Quotient of powers

This property states that when dividing two powers with the same base, we subtract the exponents.

LaTeX: \frac{x^n}{x^m}=x^{n-m}xnxm=xn−m

Examples

LaTeX: \frac{4^7}{4^3}=4^{7-3}=4^44743=47−3=44

LaTeX: \frac{x^9}{x^7}=x^{9-7}=x^2x9x7=x9−7=x2

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Power of a power property

This property states that to find a power of a power we multiply the exponents.

LaTeX: \left(x^n\right)^m=\:x^{n\cdot m}(xn)m=xn⋅m

Examples

LaTeX: \left(4^3\right)^5=4^{3\cdot5}=4^{15}(43)5=43⋅5=415

LaTeX: \left(x^3\right)^7=x^{3\cdot7}=x^{21}(x3)7=x3⋅7=x21

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Power of a product

This property states that when taking the power of a product, we multiply the powers of the factors.

LaTeX: \left(x\cdot y\right)^n=x^n\cdot y^n(x⋅y)n=xn⋅yn

Examples

 

Power of a quotient

This property states that when taking the power of a quotient, we divide the powers of the numerator and of the denominator.

LaTeX: \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}(xy)n=xnyn

Examples

LaTeX: \left(\frac{3}{4}\right)^7=\frac{3^7}{4^7}(34)7=3747

LaTeX: \left(\frac{x}{y}\right)^{2\:}=\frac{x^2}{y^2}(xy)2=x2y2