Question

1. there are 25,400,000 nanometers in an inch. what is this number written in scientific notation? a. 2.54×10^6 b. 2.54×10^7 c. 2.54×10^8 d. 2.54×10^9 3. the width of a rectangle is 2^6 units, and the length is 2^5 units. what is the area of the rectangle? a. 2^1 square units b. 2^11 square units c. 2^30 square units d. 4^11 square units 4. the width of a certain strand of human hair is about 1.5×10^(-3) cm. what is the width of 2×10^5 of these hairs placed next to each other? a. 3.5×10^8 cm b. 3×10^(-2) cm c. 3×10^8 cm d. 3×10^2 cm 5. which has the same value as 5^1 + 4^0? a. 9 b. 8 c. 6 d. 5 6. which of the following expressions is equivalent to 5(7^27^2)+5(7^47^(-4))-(7^87^(-4))? a. 4(7^4)+5 b. 6(7^8)+1 c. 10(7^4)-(7^(-2)) d. 10(7^4)-(7^2)

Explanation:

Step1: Convert 25400000 to scientific - notation

Move the decimal point 7 places to the left to get a number between 1 and 10. So, $25400000 = 2.54\times10^{7}$.

Step2: Calculate the area of the rectangle

Use the formula $A = lw$, where $l = 2^{5}$ and $w = 2^{6}$. According to the rule of exponents $a^{m}\times a^{n}=a^{m + n}$, $2^{6}\times2^{5}=2^{6 + 5}=2^{11}$.

Step3: Find the total width of the hairs

Multiply the width of one hair by the number of hairs. $(1.5\times10^{-3})\times(2\times10^{5})=(1.5\times2)\times(10^{-3}\times10^{5}) = 3\times10^{-3 + 5}=3\times10^{2}$.

Step4: Evaluate $5^{1}+4^{0}$

Since $5^{1}=5$ and $4^{0}=1$, then $5^{1}+4^{0}=5 + 1=6$.

Step5: Simplify the expression $5(7^{2}7^{2})+5(7^{4}7^{-4})-(7^{8}7^{-4})$

First, use the rule $a^{m}\times a^{n}=a^{m + n}$. $7^{2}7^{2}=7^{4}$, $7^{4}7^{-4}=7^{0}=1$, $7^{8}7^{-4}=7^{4}$. The expression becomes $5\times7^{4}+5\times1 - 7^{4}=(5 - 1)\times7^{4}+5=4\times7^{4}+5$.

Answer:

1. B. $2.54\times10^{7}$
2. B. $2^{11}$ square units
3. D. $3\times10^{2}$ cm
4. C. 6
5. A. $4(7^{4})+5$