Question

circle all the expressions that represent the same value.
$(5.4×10^{3})+(9.7×10^{3})$ $1.024×10^{4}$
$(7.234×10^{3})-(6.21×10^{2})$
$6.37×10^{2}$ $0.102400$
simplify. then compare using $<$, $>$ or $=$.

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

Explanation:

Problem 1: Simplify and Compare \((8.47 \times 10^{4}) + (9.63 \times 10^{3})\) and \(9.333 \times 10^{4}\)

Step 1: Convert to same exponent

Convert \(9.63 \times 10^{3}\) to \(0.963 \times 10^{4}\) (since \(10^{3}=10^{-1}\times10^{4}\), so \(9.63\times10^{3}=9.63\times10^{-1}\times10^{4} = 0.963\times10^{4}\))

Step 2: Add the numbers

\((8.47\times 10^{4})+(0.963\times 10^{4})=(8.47 + 0.963)\times10^{4}=9.433\times10^{4}\)

Step 3: Compare

Compare \(9.433\times10^{4}\) and \(9.333\times10^{4}\). Since \(9.433>9.333\), we have \(9.433\times10^{4}>9.333\times10^{4}\)

Step 1: Convert to same exponent

Convert \(6.43\times10^{-2}\) to \(643\times10^{-4}\) (since \(10^{-2}=100\times10^{-4}\), so \(6.43\times10^{-2}=6.43\times100\times10^{-4}=643\times10^{-4}\))

Step 2: Subtract the numbers

\((643\times10^{-4})-(6.87\times10^{-4})=(643 - 6.87)\times10^{-4}=636.13\times10^{-4}\)

Step 3: Convert to decimal

\(636.13\times10^{-4}=0.063613\)

Step 4: Compare

Compare \(0.063613\) and \(0.062\). Since \(0.063613>0.062\), we have \((6.43 \times 10^{-2})-(6.87 \times 10^{-4})>0.062\)

Step 1: Convert to same exponent

Convert \(1.04\times10^{-4}\) to \(10400\times10^{-8}\) (since \(10^{-4}=10000\times10^{-8}\), so \(1.04\times10^{-4}=1.04\times10000\times10^{-8}=10400\times10^{-8}\))

Step 2: Add the numbers

\((10400\times10^{-8})+(6.5\times10^{-8})=(10400 + 6.5)\times10^{-8}=10406.5\times10^{-8}\)

Step 3: Convert to scientific notation

\(10406.5\times10^{-8}=1.04065\times10^{-4}\) (divide \(10406.5\) by \(10000\) to get \(1.04065\) and multiply \(10^{-8}\) by \(10000 = 10^{4}\), so \(10^{-8}\times10^{4}=10^{-4}\))

Step 4: Compare

Compare \(1.04065\times10^{-4}\) and \(1.05 \times 10^{-2}\). Convert \(1.05 \times 10^{-2}\) to \(10500\times10^{-8}\) (since \(10^{-2}=1000000\times10^{-8}\)? Wait, better to convert to decimal. \(1.04065\times10^{-4}=0.000104065\) and \(1.05 \times 10^{-2}=0.0105\). Since \(0.000104065<0.0105\), we have \((1.04 \times 10^{-4})+(6.5 \times 10^{-8})<1.05 \times 10^{-2}\)

Answer:

\((8.47 \times 10^{4}) + (9.63 \times 10^{3})>9.333 \times 10^{4}\)

Problem 2: Simplify and Compare \((6.43 \times 10^{-2})-(6.87 \times 10^{-4})\) and \(0.062\)