Question

algebra 1
hw exponent properties
simplify. your answer should contain only positive exponents.

1. $6^3 \cdot 6^8$
2. $7n^5 \cdot 8nm^9$
3. $(4^3)^4$
4. $(3y)^4$
5. $\frac{2^4}{2^8}$
6. $\frac{2m^8}{7nm^4}$
7. $6^{-4}$
8. $x^4y^{-3}$
9. $(2^3 \cdot 2^0)^0$
10. $\frac{2x^{-3}}{2y^{-1} \cdot 4xy^2}$

Explanation:

Step1: Multiply same base, add exponents

$6^3 \cdot 6^8 = 6^{3+8} = 6^{11}$

Step2: Multiply coefficients, add exponents

$7n^5 \cdot 8mn^9 = (7 \cdot 8)m^{1}n^{5+9} = 56mn^{14}$

Step3: Power of a power, multiply exponents

$(4^3)^4 = 4^{3 \cdot 4} = 4^{12}$

Step4: Distribute exponent to each factor

$(3y)^4 = 3^4 y^4 = 81y^4$

Step5: Divide same base, subtract exponents

$\frac{2^4}{2^8} = 2^{4-8} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}$

Step6: Simplify variable exponents

$\frac{2m^8}{7nm^4} = \frac{2}{7n}m^{8-4} = \frac{2m^4}{7n}$

Step7: Negative exponent to positive

$6^{-4} = \frac{1}{6^4} = \frac{1}{1296}$

Step8: Move negative exponent to denominator

$x^4 y^{-3} = \frac{x^4}{y^3}$

Step9: Zero exponent rule (non-zero to 0 power=1)

$(2^3 \cdot 2^0)^0 = (8 \cdot 1)^0 = 8^0 = 1$

Step10: Simplify numerator/denominator step-by-step

First, simplify constants and like bases:
Numerator: $2x^{-3} = \frac{2}{x^3}$
Denominator: $2y^{-1} \cdot 4xy^2 = (2 \cdot 4)x^{1}y^{-1+2} = 8xy$
Combine: $\frac{\frac{2}{x^3}}{8xy} = \frac{2}{8x^4y} = \frac{1}{4x^4y}$

Answer:

1. $6^{11}$
2. $56mn^{14}$
3. $4^{12}$
4. $81y^4$
5. $\frac{1}{16}$
6. $\frac{2m^4}{7n}$
7. $\frac{1}{1296}$
8. $\frac{x^4}{y^3}$
9. $1$
10. $\frac{1}{4x^4y}$