The Rules for Exponents
Learning Outcomes
- Recall the properties of exponents and use them to rewrite expressions containing exponents.
The Product Rule of Exponents
For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that[latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
The Quotient Rule of Exponents
For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that[latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
The Power Rule of Exponents
For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that[latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
The Zero Exponent Rule of Exponents
For any nonzero real number [latex]a[/latex], the zero exponent rule of exponents states that[latex]{a}^{0}=1[/latex]
The Negative Rule of Exponents
For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that[latex]{a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}[/latex]
The Power of a Product Rule of Exponents
For any real numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]n[/latex], the power of a product rule of exponents states that[latex]\large{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex]
The Power of a Quotient Rule of Exponents
For any real numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]n[/latex], the power of a quotient rule of exponents states that[latex]\large{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}[/latex]
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