Question

evaluate. box the final answer.
27.

1. ((4j^{6})^{4})
2. ((3v^{9})^{5})

30.

1. ((2m^{5}n^{8})^{7})
2. ((-3w^{2}z^{9})^{5})
3. (4^{-6})
4. ((-5)^{-3})

simplify. express answer using positive exponents.

1. (9 \bullet 9 \bullet x \bullet w \bullet x \bullet y \bullet w \bullet 9 \bullet y \bullet y)
2. ((3x^{2}y^{6}z)^{3}(15xyz))

Explanation:

Step1: Evaluate exponents first

$8^2=64$, $4^2=16$

Step2: Subtract the results

$64 - 16 = 48$

Step1: Apply power of a product rule

$(4j^6)^4 = 4^4 \cdot (j^6)^4$

Step2: Calculate constants and exponents

$4^4=256$, $(j^6)^4=j^{6 \times 4}=j^{24}$

Step3: Combine terms

$256j^{24}$

Step1: Apply power of a product rule

$(3v^9)^5 = 3^5 \cdot (v^9)^5$

Step2: Calculate constants and exponents

$3^5=243$, $(v^9)^5=v^{9 \times 5}=v^{45}$

Step3: Combine terms

$243v^{45}$

Step1: Simplify numerator exponents

$3^{10} \cdot (-6)^8$; denominator: $3^2 \cdot (-6)^2$

Step2: Apply quotient of powers rule

$3^{10-2} \cdot (-6)^{8-2} = 3^8 \cdot (-6)^6$

Step3: Calculate constants

$3^8=6561$, $(-6)^6=46656$

Step4: Multiply results

$6561 \times 46656 = 306674688$

Step1: Apply power of a product rule

$(2m^5n^8)^7 = 2^7 \cdot (m^5)^7 \cdot (n^8)^7$

Step2: Calculate constants and exponents

$2^7=128$, $(m^5)^7=m^{35}$, $(n^8)^7=n^{56}$

Step3: Combine terms

$128m^{35}n^{56}$

Step1: Apply power of a product rule

$(-3w^2z^9)^5 = (-3)^5 \cdot (w^2)^5 \cdot (z^9)^5$

Step2: Calculate constants and exponents

$(-3)^5=-243$, $(w^2)^5=w^{10}$, $(z^9)^5=z^{45}$

Step3: Combine terms

$-243w^{10}z^{45}$

Step1: Rewrite negative exponent as reciprocal

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

Step2: Calculate positive exponent

$4^6=4096$

Step3: Final form

$\frac{1}{4096}$

Step1: Rewrite negative exponent as reciprocal

$(-5)^{-3} = \frac{1}{(-5)^3}$

Step2: Calculate positive exponent

$(-5)^3=-125$

Step3: Final form

$-\frac{1}{125}$

Step1: Group like bases and constants

$9 \cdot 9 \cdot x \cdot x \cdot w \cdot w \cdot y \cdot y \cdot y$

Step2: Apply product of powers rule

$9^2 \cdot x^2 \cdot w^2 \cdot y^3$

Step3: Calculate constant

$9^2=81$

Step4: Combine terms

$81x^2w^2y^3$

Step1: Apply power of a product rule

$(3x^2y^6z)^3 = 3^3 \cdot (x^2)^3 \cdot (y^6)^3 \cdot z^3$

Step2: Calculate constants and exponents

$3^3=27$, $(x^2)^3=x^6$, $(y^6)^3=y^{18}$, $z^3=z^3$

Step3: Multiply with remaining term

$27x^6y^{18}z^3 \cdot 15xyz$

Step4: Combine like terms

$27 \times 15 \cdot x^{6+1} \cdot y^{18+1} \cdot z^{3+1}$

Step5: Calculate final values

$405x^7y^{19}z^4$

Answer:

1. $48$
2. $256j^{24}$
3. $243v^{45}$
4. $306674688$
5. $128m^{35}n^{56}$
6. $-243w^{10}z^{45}$
7. $\frac{1}{4096}$
8. $-\frac{1}{125}$
9. $81x^2w^2y^3$
10. $405x^7y^{19}z^4$