Product of Powers
When multiplying powers with the same base, keep the ______ and ______ the exponents.
When multiplying powers with the same base, keep the base and add the exponents.
When dividing powers with the same nonzero base, keep the base and ______ the denominator’s exponent from the numerator’s exponent.
When dividing powers with the same nonzero base, keep the base and subtract the denominator’s exponent from the numerator’s exponent.
When raising a product to a power, apply the exponent to ______ factor. When raising a quotient to a power, apply it to both the ______ and the ______.
When raising a product to a power, apply the exponent to each factor. When raising a quotient to a power, apply it to both the numerator and the denominator.
When raising a power to another power, keep the ______ and ______ the exponents.
When raising a power to another power, keep the base and multiply the exponents.
Any nonzero number raised to the zero power equals ______. Any number raised to the first power equals ______.
Any nonzero number raised to the zero power equals 1. Any number raised to the first power equals itself.
Write as a single power: 6^3\cdot6^4
\(6^7\)
Write as a single power: \frac{8^7}{8^3}.
8^4
Rewrite \(3\cdot4)^2 as a product of two powers.
3^2\cdot4^2
Write as a single power: (10^3)^2.
10^6
Evaluate both: 19^0 and 19^1.
19^0=1
19^1=19
Write as a single power: (\frac{2}{3})^2\cdot(\frac{2}{3})^5\cdot(\frac{2}{3})
(\frac{2}{3})^8
Write as a single power: \frac{(-3)^9}{(-3)^5}.
(-3)^4
Rewrite using the power of a quotient law, then evaluate: \(frac{-2}{3})^3.
\frac{(-2)^3}{3^3}=\frac{-8}{27}
Find the missing exponent: (5^3)^{\square}=5^{12}.
4 because 3\cdot4=12.
Evaluate both: (-\frac{3}{4})^0 and (-\frac{3}{4})^1.
(-\frac{3}{4})^0=1
(-\frac{3}{4})^1=-\frac{3}{4}
A student writes 7^3\cdot7^2=7^6.
Explain the mistake and correct the answer.
They multiplied the exponents instead of adding. The correct answer is 7^{(3+2)}=7^5.
Simplify to a single power using exponent laws: \frac{(5^7*5^2)}{5^6}.
Explain which laws you used to simplify.
5^{(7+2-6)}=5^3. Product of powers, quotient of powers.
A student writes 2^3\cdot5^3=10^3. Is the student correct? Why or why not?
Yes, the student is correct. When multiplying powers with the same exponent, multiply the bases and keep the exponent:
2^3\cdot5^3=(2\cdot5)^3=10^3
Write as a single power, then evaluate: ((-2)^3)^2.
(-2)^6=64
A student says 8^0+8^1=8^1=8 because “adding zero does not change the exponent.” Explain the mistake and evaluate correctly.
The product rule applies to multiplication, not addition.
8^0+8^1=1+8=9.
Simplify to a single power using exponent laws: \frac{(2^3)^2\cdot2^4}{2^5}.
Explain which laws you used.
2^{(6+4-5)}=2^5
Power of a power, product of powers, quotient of powers.
Simplify \frac{3^2}{3^5} using the quotient of powers law.
a. What is the simplified exponential expression?
b. Expand the powers and cancel common factors. What do you notice about the exponent from part A and the remaining fraction?
a. \frac{3^2}{3^5}=3^{(2-5)}=3^{-3}
b. \frac{3\cdot3}{3\cdot3\cdot3\cdot3\cdot3}=\frac{1}{3\cdot3\cdot3}=\frac{1}{3^3}
The exponent -3 corresponds to three factors of 3 remaining in the denominator. Therefore, 3^-3=\frac{1}{3^3}: a negative exponent tells us to take the reciprocal.
Simplify to a single power using exponent laws: \frac{(-12)^3}{3^3}\cdot(-4)^2.
Explain which laws you used.
(\frac{-12}{3})^3\cdot(-4)^2=(-4)^3\cdot(-4)^2=(-4)^5
Power of a quotient, product of powers.
Simplify using exponent laws, then evaluate: \frac{\((2^2)^3)^2}{2^8}.
Explain which laws you used.
2^{(2\cdot3\cdot2)-8}=2^4=16
Power of a power, quotient of powers.
Simplify using exponent laws, then evaluate: \frac{0.5^4}{0.5^3}\cdot(8^2)^0.
Explain which laws you used.
\frac{0.5^4}{0.5^3}\cdot(8^2)^0=0.5^{(4-3)}*8^{(2*0)}=0.5*1=0.5
Quotient of powers, identity exponent, zero exponent.