Question

circle all the expressions that represent the same value.
$(5.4 \times 10^{-3}) + (9.7 \times 10^{-2})$ $1.024 \times 10^{-1}$
$(7.234 \times 10^{-1}) - (6.21 \times 10^{-2})$
$6.37 \times 10^{-2}$ $0.102400$
simplify. then compare using $<$, $>$ or $=$.

1. $(8.47 \times 10^4) + (9.63 \times 10^3)$ $\square$ $9.333 \times 10^4$
2. $(6.43 \times 10^{-2}) - (6.87 \times 10^{-4})$ $\square$ $0.062$
3. $(1.04 \times 10^{-4}) + (6.5 \times 10^{-6})$ $\square$ $1.05 \times 10^{-1}$

Explanation:

Part 1: Circle Expressions with Same Value

Step 1: Simplify \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\)

Convert to same exponent:
\(5.4 \times 10^{-3} = 0.54 \times 10^{-2}\)
Add: \(0.54 \times 10^{-2} + 9.7 \times 10^{-2} = (0.54 + 9.7) \times 10^{-2} = 10.24 \times 10^{-2} = 1.024 \times 10^{-1}\) (since \(10.24 \times 10^{-2} = 1.024 \times 10^{-1}\))

Step 2: Simplify \(1.024 \times 10^{-1}\)

\(1.024 \times 10^{-1} = 0.1024\), which matches \(0.102400\).

Step 3: Simplify \((7.234 \times 10^{-1}) - (6.21 \times 10^{-2})\)

Convert to same exponent:
\(7.234 \times 10^{-1} = 72.34 \times 10^{-2}\)
Subtract: \(72.34 \times 10^{-2} - 6.21 \times 10^{-2} = (72.34 - 6.21) \times 10^{-2} = 66.13 \times 10^{-2} = 6.613 \times 10^{-1}\) (Wait, no—wait, \(7.234 \times 10^{-1} = 0.7234\), \(6.21 \times 10^{-2} = 0.0621\). Subtract: \(0.7234 - 0.0621 = 0.6613 = 6.613 \times 10^{-1}\)? Wait, no, the original problem has \(6.37 \times 10^{-2}\)—maybe a typo? Wait, no, let's recheck. Wait, the first expression: \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\) = \(0.0054 + 0.097 = 0.1024\), which is \(1.024 \times 10^{-1}\) and \(0.102400\). So these three: \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\), \(1.024 \times 10^{-1}\), \(0.102400\) are equal.

Part 2: Simplify and Compare

1. \((8.47 \times 10^4) + (9.63 \times 10^3)\) vs \(9.333 \times 10^4\)

Step 1: Convert \(9.63 \times 10^3\) to \(10^4\): \(9.63 \times 10^3 = 0.963 \times 10^4\)
Step 2: Add: \(8.47 \times 10^4 + 0.963 \times 10^4 = (8.47 + 0.963) \times 10^4 = 9.433 \times 10^4\)
Step 3: Compare \(9.433 \times 10^4\) vs \(9.333 \times 10^4\): \(9.433 > 9.333\), so \(>\)

2. \((6.43 \times 10^{-2}) - (6.87 \times 10^{-4})\) vs \(0.062\)

Step 1: Convert to same exponent: \(6.43 \times 10^{-2} = 643 \times 10^{-4}\), \(6.87 \times 10^{-4}\)
Step 2: Subtract: \(643 \times 10^{-4} - 6.87 \times 10^{-4} = (643 - 6.87) \times 10^{-4} = 636.13 \times 10^{-4} = 0.063613\)
Step 3: Compare \(0.063613\) vs \(0.062\): \(0.063613 > 0.062\), so \(>\)

3. \((1.04 \times 10^{-4}) + (6.5 \times 10^{-6})\) vs \(1.05 \times 10^{-4}\)

Step 1: Convert \(6.5 \times 10^{-6}\) to \(10^{-4}\): \(6.5 \times 10^{-6} = 0.065 \times 10^{-4}\)
Step 2: Add: \(1.04 \times 10^{-4} + 0.065 \times 10^{-4} = (1.04 + 0.065) \times 10^{-4} = 1.105 \times 10^{-4}\)
Step 3: Compare \(1.105 \times 10^{-4}\) vs \(1.05 \times 10^{-4}\): \(1.105 > 1.05\), so \(>\)

Final Answers (Part 1: Circle)

• \(\boldsymbol{(5.4 \times 10^{-3}) + (9.7 \times 10^{-2})}\)
• \(\boldsymbol{1.024 \times 10^{-1}}\)
• \(\boldsymbol{0.102400}\)

Final Answers (Part 2: Compare)

1. \(\boldsymbol{>}\)
2. \(\boldsymbol{>}\)
3. \(\boldsymbol{>}\)

Answer:

Part 1: Circle Expressions with Same Value

Step 1: Simplify \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\)

Convert to same exponent:
\(5.4 \times 10^{-3} = 0.54 \times 10^{-2}\)
Add: \(0.54 \times 10^{-2} + 9.7 \times 10^{-2} = (0.54 + 9.7) \times 10^{-2} = 10.24 \times 10^{-2} = 1.024 \times 10^{-1}\) (since \(10.24 \times 10^{-2} = 1.024 \times 10^{-1}\))

Step 2: Simplify \(1.024 \times 10^{-1}\)

\(1.024 \times 10^{-1} = 0.1024\), which matches \(0.102400\).

Step 3: Simplify \((7.234 \times 10^{-1}) - (6.21 \times 10^{-2})\)

Convert to same exponent:
\(7.234 \times 10^{-1} = 72.34 \times 10^{-2}\)
Subtract: \(72.34 \times 10^{-2} - 6.21 \times 10^{-2} = (72.34 - 6.21) \times 10^{-2} = 66.13 \times 10^{-2} = 6.613 \times 10^{-1}\) (Wait, no—wait, \(7.234 \times 10^{-1} = 0.7234\), \(6.21 \times 10^{-2} = 0.0621\). Subtract: \(0.7234 - 0.0621 = 0.6613 = 6.613 \times 10^{-1}\)? Wait, no, the original problem has \(6.37 \times 10^{-2}\)—maybe a typo? Wait, no, let's recheck. Wait, the first expression: \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\) = \(0.0054 + 0.097 = 0.1024\), which is \(1.024 \times 10^{-1}\) and \(0.102400\). So these three: \((5.4 \times 10^{-3}) + (9.7 \times 10^{-2})\), \(1.024 \times 10^{-1}\), \(0.102400\) are equal.

Part 2: Simplify and Compare

1. \((8.47 \times 10^4) + (9.63 \times 10^3)\) vs \(9.333 \times 10^4\)

Step 1: Convert \(9.63 \times 10^3\) to \(10^4\): \(9.63 \times 10^3 = 0.963 \times 10^4\)
Step 2: Add: \(8.47 \times 10^4 + 0.963 \times 10^4 = (8.47 + 0.963) \times 10^4 = 9.433 \times 10^4\)
Step 3: Compare \(9.433 \times 10^4\) vs \(9.333 \times 10^4\): \(9.433 > 9.333\), so \(>\)

2. \((6.43 \times 10^{-2}) - (6.87 \times 10^{-4})\) vs \(0.062\)

Step 1: Convert to same exponent: \(6.43 \times 10^{-2} = 643 \times 10^{-4}\), \(6.87 \times 10^{-4}\)
Step 2: Subtract: \(643 \times 10^{-4} - 6.87 \times 10^{-4} = (643 - 6.87) \times 10^{-4} = 636.13 \times 10^{-4} = 0.063613\)
Step 3: Compare \(0.063613\) vs \(0.062\): \(0.063613 > 0.062\), so \(>\)

3. \((1.04 \times 10^{-4}) + (6.5 \times 10^{-6})\) vs \(1.05 \times 10^{-4}\)

Step 1: Convert \(6.5 \times 10^{-6}\) to \(10^{-4}\): \(6.5 \times 10^{-6} = 0.065 \times 10^{-4}\)
Step 2: Add: \(1.04 \times 10^{-4} + 0.065 \times 10^{-4} = (1.04 + 0.065) \times 10^{-4} = 1.105 \times 10^{-4}\)
Step 3: Compare \(1.105 \times 10^{-4}\) vs \(1.05 \times 10^{-4}\): \(1.105 > 1.05\), so \(>\)

Final Answers (Part 1: Circle)

• \(\boldsymbol{(5.4 \times 10^{-3}) + (9.7 \times 10^{-2})}\)
• \(\boldsymbol{1.024 \times 10^{-1}}\)
• \(\boldsymbol{0.102400}\)

Final Answers (Part 2: Compare)

1. \(\boldsymbol{>}\)
2. \(\boldsymbol{>}\)
3. \(\boldsymbol{>}\)