Question

4 a cube - shaped paperweight has a volume of 216 cm³. what is the length of the paperweight? a 4 cm b 5 cm c 6 cm d 8 cm 5 look at the equations below. tell whether each equation is true or false. a. (4⁰)⁶ = 4⁶ true false b. $\frac{3^{8}}{3^{-2}}=3^{10}$ true false c. $2^{-5}\times2^{3}=\frac{1}{2^{2}}$ true false d. $6^{4}\times3^{4}=18^{8}$ true false

Explanation:

Step1: Recall volume formula for cube

The volume formula for a cube is $V = s^{3}$, where $s$ is the side - length of the cube. Given $V = 216\ cm^{3}$, we need to solve for $s$.

Step2: Solve for side - length

We have the equation $s^{3}=216$. Taking the cube - root of both sides, $s=\sqrt[3]{216}$. Since $6\times6\times6 = 216$, $s = 6\ cm$.

Step3: Analyze equation (a)

For $(4^{0})^{6}$, by the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, and $4^{0}=1$, so $(4^{0})^{6}=1^{6}=1$, while $4^{6}=4096$. So $(4^{0})^{6}=4^{6}$ is False.

Step4: Analyze equation (b)

For $\frac{3^{8}}{3^{- 2}}$, by the quotient rule $\frac{a^{m}}{a^{n}}=a^{m - n}$, we have $\frac{3^{8}}{3^{-2}}=3^{8-(-2)}=3^{10}$. So $\frac{3^{8}}{3^{-2}}=3^{10}$ is True.

Step5: Analyze equation (c)

For $2^{-5}\times2^{3}$, by the product rule $a^{m}\times a^{n}=a^{m + n}$, we have $2^{-5}\times2^{3}=2^{-5 + 3}=2^{-2}=\frac{1}{2^{2}}$. So $2^{-5}\times2^{3}=\frac{1}{2^{2}}$ is True.

Step6: Analyze equation (d)

For $6^{4}\times3^{4}$, by the product - of - powers rule $a^{m}\times b^{m}=(a\times b)^{m}$, we have $6^{4}\times3^{4}=(6\times3)^{4}=18^{4}$, not $18^{8}$. So $6^{4}\times3^{4}=18^{8}$ is False.

Answer:

1. C. 6 cm
2. a. False

b. True
c. True
d. False