The Power of a Product Rule of Exponents
For any real numbers
a and
b and any integer
n, the power of a product rule of exponents states that
(ab)n=anbn
Example
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
- (ab2)3
- (2at)15
- (−2w3)3
- (−7z)41
- (e−2f2)7
Answer:
Use the product and quotient rules and the new definitions to simplify each expression.
- (ab2)3=(a)3⋅(b2)3=a1⋅3⋅b2⋅3=a3b6
- (2at)15=(2a)15⋅(t)15=2a⋅15⋅t15=215a⋅t15
- (−2w3)3=(−2)3⋅(w3)3=−8⋅w3⋅3=−8w9
- (−7z)41=(−7)4⋅(z)41=2,401z41
- (e−2f2)7=(e−2)7⋅(f2)7=e−2⋅7⋅f2⋅7=e−14f14=e14f14
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Caution! Do not try to apply this rule to sums.
Think about the expression
(2+3)2
Does (2+3)2 equal 22+32?
No, it does not because of the order of operations!
(2+3)2=52=25
and
22+32=4+9=13
Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied or divided.
The Power of a Quotient Rule of Exponents
For any real numbers
a and
b and any integer
n, the power of a quotient rule of exponents states that
(ba)n=bnan
Example
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
- (z114)3
- (q3p)6
- (t2−1)27
- (j3k−2)4
- (m−2n−2)3
Answer:
- (z114)3=(z11)3(4)3=z11⋅364=z3364
- (q3p)6=(q3)6(p)6=q3⋅6p1⋅6=q18p6
- (t2−1)27=(t2)27(−1)27=t2⋅27−1=t54−1=−t541
- (j3k−2)4=(k2j3)4=(k2)4(j3)4=k2⋅4j3⋅4=k8j12
- (m−2n−2)3=(m2n21)3=(m2n2)3(1)3=(m2)3(n2)31=m2⋅3⋅n2⋅31=m6n61
The following video provides more examples of simplifying expressions using the power of a quotient and other exponent rules.
https://youtu.be/BoBe31pRxFM
Summary
- Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.
- The product rule for exponents: For any number x and any integers a and b, (xa)(xb)=xa+b.
- The quotient rule for exponents: For any non-zero number x and any integers a and b, xbxa=xa−b
- The power rule for exponents:
- For any nonzero numbers a and b and any integer n, (ab)n=an⋅bn.
- For any number a, any non-zero number b, and any integer n, (ba)n=bnan