Question

determine the value of the missing exponent.
1 ( 4.725 \times 10^{10} + 8.6 \times 10^{square} = 4.811 \times 10^{10} )
a. 8 b. 10 c. 12
2 ( 1.04 \times 10^{8} - 9.2 \times 10^{square} = 1.2 \times 10^{7} )
a. 6 b. 7 c. 8
3 ( 42 \text{ million} + 9.63 \times 10^{square} = 1.005 \times 10^{9} )
a. 6 b. 7 c. 8

Explanation:

Problem 1

Step1: Rearrange the equation

Let \( y = 10^{\square} \), then the equation \( 4.725\times10^{10}+8.6\times y = 4.811\times10^{10} \) can be rearranged to \( 8.6\times y=4.811\times 10^{10}-4.725\times 10^{10} \)

Step2: Calculate the right - hand side

\( 4.811\times 10^{10}-4.725\times 10^{10}=(4.811 - 4.725)\times10^{10}=0.086\times10^{10} \)

Step3: Solve for y

From \( 8.6\times y = 0.086\times10^{10} \), we get \( y=\frac{0.086\times 10^{10}}{8.6} \)
\( \frac{0.086\times 10^{10}}{8.6}=\frac{8.6\times 10^{8}}{8.6}=10^{8} \)
Since \( y = 10^{\square}=10^{8} \), the missing exponent \( \square = 8 \)

Step1: Rearrange the equation

Let \( z = 10^{\square} \), the equation \( 1.04\times10^{8}-9.2\times z = 1.2\times10^{7} \) can be rearranged to \( 9.2\times z=1.04\times 10^{8}-1.2\times 10^{7} \)

Step2: Convert to the same power of 10

\( 1.04\times 10^{8}=10.4\times 10^{7} \), so \( 1.04\times 10^{8}-1.2\times 10^{7}=10.4\times 10^{7}-1.2\times 10^{7} \)

Step3: Calculate the right - hand side

\( 10.4\times 10^{7}-1.2\times 10^{7}=(10.4 - 1.2)\times10^{7}=9.2\times 10^{7} \)

Step4: Solve for z

From \( 9.2\times z = 9.2\times 10^{7} \), we get \( z = 10^{7} \)
Since \( z = 10^{\square}=10^{7} \), the missing exponent \( \square=7 \)

Step1: Convert 42 million to scientific notation

42 million \( = 42\times10^{6}=4.2\times 10^{7} \), and \( 1.005\times 10^{9}=100.5\times 10^{7} \)
Let \( w = 10^{\square} \), the equation \( 4.2\times 10^{7}+9.63\times w=100.5\times 10^{7} \) can be rearranged to \( 9.63\times w=100.5\times 10^{7}-4.2\times 10^{7} \)

Step2: Calculate the right - hand side

\( 100.5\times 10^{7}-4.2\times 10^{7}=(100.5 - 4.2)\times10^{7}=96.3\times 10^{7} \)

Step3: Solve for w

From \( 9.63\times w = 96.3\times 10^{7} \), we get \( w=\frac{96.3\times 10^{7}}{9.63} \)
\( \frac{96.3\times 10^{7}}{9.63}=\frac{9.63\times 10^{8}}{9.63}=10^{8} \)
Since \( w = 10^{\square}=10^{8} \), the missing exponent \( \square = 8 \)

Answer:

a. 8

Problem 2