\(0.06 \times 10^2 =\)
Digits shift left 2 places
Problem 5
\(4.125 \times 10^3 =\)
Digits shift left 3 places
Problem 6
\(16.8 \times 10^2 =\)
Digits shift left 2 places
5
\(52 \times 10^{\mathbf{n}} = 5,200\)
\(\mathbf{n} =\)
\(9 \times 10^{\mathbf{n}} = 90,000\)
\(\mathbf{n} =\)
\(6.4 \times 10^{\mathbf{n}} = 640\)
\(\mathbf{n} =\)
\(0.8 \times 10^{\mathbf{n}} = 800\)
\(\mathbf{n} =\)
6
An astronomer calculates that the average distance from Earth to the Moon is approximately \(3.84 \times 10^5\) kilometers. What is this distance written in standard numerical form?
Show Your Step-by-Step Thinking / Shifts:
Standard Form Distance: km
7
Marcus states: "Multiplying by \(10^2\) just means sticking two zeros at the end of whatever number you have. Therefore, \(4.5 \times 10^2 = 4.500\)!"
Explain why Marcus's rule fails:
Show the correct place value shift & product:
Correct Answer: _______
Power Shift Mastery • Grade 5 Math Page 2 of 2
305,000
Middle zero preserved
F. Challenge
\(60 \times 10^4 =\)
600,000
\(60 \times 10,000 = 5\) total zeros
Power Shift Mastery • Teacher Answer Key Page 1 of 2
Answer Key: Decimals, Missing Exponents & Critique
Page 2 Solutions
4
6 pts (1 pt each)
Problem 1
\(3.4 \times 10^1 =\)
34
Decimal shifts 1 place right
Problem 2
\(5.82 \times 10^2 =\)
582
Decimal shifts 2 places right
Problem 3
\(0.7 \times 10^3 =\)
700
Shifts 1 place to 7, then +2 zeros
Problem 4
\(0.06 \times 10^2 =\)
6
Decimal shifts 2 places past 0 and 6
Problem 5
\(4.125 \times 10^3 =\)
4,125
Shifts past tenths, hundredths, thousandths
Problem 6
\(16.8 \times 10^2 =\)
1,680
Shifts 1 place to 168, then +1 zero
5
4 pts (1 pt each)
\(52 \times 10^{\mathbf{n}} = 5,200\)
\(\mathbf{n} =\) 2
\(9 \times 10^{\mathbf{n}} = 90,000\)
\(\mathbf{n} =\) 4
\(6.4 \times 10^{\mathbf{n}} = 640\)
\(\mathbf{n} =\) 2
\(0.8 \times 10^{\mathbf{n}} = 800\)
\(\mathbf{n} =\) 3
6
2 pts
Question: Average distance from Earth to Moon is \(3.84 \times 10^5\) km.
Step-by-Step Shift Demonstration:
\(10^5 = 100,000\) (5 factors of 10) • Digits shift 5 place values left (decimal hops 5 places right):
\(3.84 \xrightarrow{\times 10} 38.4 \xrightarrow{\times 10} 384 \xrightarrow{\times 10} 3,840 \xrightarrow{\times 10} 38,400 \xrightarrow{\times 10} 384,000\)
Standard Form Distance: 384,000 km
7
3 pts
Why Marcus's Rule Fails:
Adding trailing zeros after a decimal point creates an equivalent decimal, not a number 100 times larger!
\(4.5 = 4.50 = 4.500\) (the value has not increased at all).
Correct Shift & Product:
Multiplying by \(10^2\) (or 100) shifts each digit 2 place values left: the 4 ones become 4 hundreds, and the 5 tenths become 5 tens.
Correct Answer: 450
Power Shift Mastery • Teacher Answer Key Page 2 of 2
