Question

$$\frac { 1.075 \times 10 ^ { p } } { 2.5 \times 10 ^ { - 2 } } = 4.3 \times 10 ^ { 4 }$$
$$( 1.075 \div 2.5 ) \times ( 10 ^ { p } \div 10 ^ { - 2 } ) = 4.3 \times 10 ^ { 4 }$$
$$0.43 \times 10 ^ { p - ( - 2 ) } = 4.3 \times 10 ^ { 4 }$$
$$0.43 \times 10 ^ { p - ( - 2 ) - 1 } = 4.3 \times 10 ^ { 4 }$$
$$4.3 \times 10 ^ { p - 3 } = 4.3 \times 10 ^ { 4 }$$
$$p - 3 = 4$$
$$p = 7$$
circle the first step with the error. then find the correct value of themissing exponent below.
$$\frac { 1.075 \times 10 ^ { p } } { 2.5 \times 10 ^ { - 2 } } = 4.3 \times 10 ^ { 4 }$$

Explanation:

Step1: Analyze the division of coefficients and exponents

When dividing numbers in scientific notation \(\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}\). Here \(a = 1.075\), \(b=2.5\), \(m = p\), \(n=- 2\). The correct division of coefficients is \(\frac{1.075}{2.5}=0.43\), and for exponents \(10^{p}\div10^{-2}=10^{p-(-2)}=10^{p + 2}\). The error - prone step is \(0.43\times10^{p-2}\) (it should be \(0.43\times10^{p + 2}\)).

Step2: Rewrite \(0.43\) in scientific notation

Since \(0.43=4.3\times10^{-1}\), then \(0.43\times10^{p + 2}=(4.3\times10^{-1})\times10^{p + 2}\). Using the rule \(a^{m}\times a^{n}=a^{m + n}\), we get \(4.3\times10^{-1+(p + 2)}=4.3\times10^{p+1}\).

Step3: Solve for \(p\)

We know that \(4.3\times10^{p + 1}=4.3\times10^{4}\). Since the coefficients (\(4.3\)) are equal, then the exponents must be equal. So \(p+1 = 4\).

Step4: Isolate \(p\)

Subtract \(1\) from both sides of the equation \(p+1 = 4\). We get \(p=4 - 1\).

Answer:

The first step with the error is \(0.43\times10^{p - 2}\). The correct value of \(p\) is \(3\).