Basic Arithmetic 825432

1. **Find the median of the data set $12$, $15$, $15$, $18$, $20$, $22$, $24$.** The median is the middle value when the data is ordered. The data is already ordered: $12$, $15$, $15$, $18$, $20$, $22$, $24$. Since there are 7 numbers (odd count), the median is the 4th number. **Answer:** $\boxed{18}$ 2. **Find the mean of the numbers $8$, $12$, $15$, $9$, and $16$.** Formula for mean: $$\text{mean} = \frac{\text{sum of all numbers}}{\text{number of numbers}}$$ Sum: $8 + 12 + 15 + 9 + 16 = 60$ Count: $5$ Mean: $$\frac{60}{5} = 12$$ **Answer:** $\boxed{12}$ 3. **Find the mode of the data set $5$, $8$, $3$, $8$, $6$, $9$, $5$, $8$, $2$.** Mode is the number that appears most frequently. Count occurrences: - $5$ appears 2 times - $8$ appears 3 times - $3$, $6$, $9$, $2$ appear once each Mode is $8$ because it appears most. **Answer:** $\boxed{8}$ 4. **Simplify $\frac{2^2}{2^3}$.** Use the rule: $$\frac{a^m}{a^n} = a^{m-n}$$ $$\frac{2^2}{2^3} = 2^{2-3} = 2^{-1}$$ Simplify $2^{-1}$: $$2^{-1} = \frac{1}{2}$$ **Answer:** $\boxed{\frac{1}{2}}$ 5. **Simplify $\frac{3r^3}{2r}$.** Divide coefficients and variables separately: $$\frac{3}{2} \cdot \frac{r^3}{r^1} = \frac{3}{2} r^{3-1} = \frac{3}{2} r^2$$ **Answer:** $\boxed{\frac{3}{2} r^2}$ 6. **Simplify $\frac{10p^4}{6p}$.** Divide coefficients: $$\frac{10}{6} = \frac{5}{3}$$ Divide variables: $$p^{4-1} = p^3$$ So: $$\frac{10p^4}{6p} = \frac{5}{3} p^3$$ **Answer:** $\boxed{\frac{5}{3} p^3}$ 7. **Simplify $\frac{8m^3}{10m^3}$.** Divide coefficients: $$\frac{8}{10} = \frac{4}{5}$$ Divide variables: $$m^{3-3} = m^0 = 1$$ So: $$\frac{8m^3}{10m^3} = \frac{4}{5}$$ **Answer:** $\boxed{\frac{4}{5}}$ 8. **Simplify $4^2 \cdot 4^2$.** Use the rule: $$a^m \cdot a^n = a^{m+n}$$ $$4^2 \cdot 4^2 = 4^{2+2} = 4^4$$ Calculate $4^4$: $$4^4 = 4 \times 4 \times 4 \times 4 = 256$$ **Answer:** $\boxed{256}$ 9. **Simplify $4 \cdot 4^2$.** Rewrite $4$ as $4^1$: $$4^1 \cdot 4^2 = 4^{1+2} = 4^3$$ Calculate $4^3$: $$4^3 = 4 \times 4 \times 4 = 64$$ **Answer:** $\boxed{64}$ 10. **Simplify $2n^4 \cdot 5n^4$.** Multiply coefficients: $$2 \times 5 = 10$$ Multiply variables: $$n^4 \cdot n^4 = n^{4+4} = n^8$$ So: $$2n^4 \cdot 5n^4 = 10 n^8$$ **Answer:** $\boxed{10 n^8}$ 11. **Simplify $6r \cdot 5r^2$.** Multiply coefficients: $$6 \times 5 = 30$$ Multiply variables: $$r^1 \cdot r^2 = r^{1+2} = r^3$$ So: $$6r \cdot 5r^2 = 30 r^3$$ **Answer:** $\boxed{30 r^3}$ 12. **Simplify $2n^4 \cdot 6n^4$.** Multiply coefficients: $$2 \times 6 = 12$$ Multiply variables: $$n^4 \cdot n^4 = n^{8}$$ So: $$2n^4 \cdot 6n^4 = 12 n^8$$ **Answer:** $\boxed{12 n^8}$ 13. **Simplify $(3a^2)^3$.** Use the rule: $$(ab)^n = a^n b^n$$ $$(3a^2)^3 = 3^3 (a^2)^3 = 27 a^{2 \times 3} = 27 a^6$$ **Answer:** $\boxed{27 a^6}$ 14. **Simplify $(2n^4)^4$.** $$(2n^4)^4 = 2^4 (n^4)^4 = 16 n^{16}$$ **Answer:** $\boxed{16 n^{16}}$ 15. **Simplify $(3x^4)^4$.** $$(3x^4)^4 = 3^4 (x^4)^4 = 81 x^{16}$$ **Answer:** $\boxed{81 x^{16}}$ 16. **Simplify $2^2$.** $$2^2 = 4$$ **Answer:** $\boxed{4}$ 17. **Simplify $2^1$.** $$2^1 = 2$$ **Answer:** $\boxed{2}$ 18. **Simplify $2^0$.** Any nonzero number to the zero power is 1: $$2^0 = 1$$ **Answer:** $\boxed{1}$ 19. **Simplify $2^{-1}$.** Negative exponent means reciprocal: $$2^{-1} = \frac{1}{2}$$ **Answer:** $\boxed{\frac{1}{2}}$ 20. **Simplify $2^{-2}$.** $$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$$ **Answer:** $\boxed{\frac{1}{4}}$