Question

1. national debt in 1915, the u.s. national 3.1×10⁹ dollars. in 2015, the u.s. national 5.9×10³ times more than in 1915. what wa 10. simplify each expression. assume that no denom (\frac{a^{7}b^{9}}{ab^{5}})

Explanation:

Problem 9 (National Debt)

Step 1: Identify the debt in 1915 and the multiple

The debt in 1915 is \(3.1\times10^{9}\) dollars, and in 2015 it is \(5.9\times10^{3}\) times more than in 1915. So we need to multiply these two numbers: \((3.1\times10^{9})\times(5.9\times10^{3})\)

Step 2: Multiply the coefficients and the powers of 10 separately

First, multiply the coefficients: \(3.1\times5.9 = 18.29\)

Then, multiply the powers of 10 using the rule \(a^{m}\times a^{n}=a^{m + n}\): \(10^{9}\times10^{3}=10^{9 + 3}=10^{12}\)

Step 3: Combine the results

Now, combine the coefficient and the power of 10: \(18.29\times10^{12}\)

But we should write this in proper scientific notation. Since \(18.29 = 1.829\times10^{1}\), we have:

\(1.829\times10^{1}\times10^{12}=1.829\times10^{13}\) (using the rule \(a^{m}\times a^{n}=a^{m + n}\) again for the powers of 10)

Step 1: Use the quotient rule for exponents (\(\frac{a^{m}}{a^{n}}=a^{m - n}\)) for \(a\) and \(b\) separately

For the variable \(a\): \(\frac{a^{7}}{a^{1}}=a^{7 - 1}=a^{6}\) (since \(a = a^{1}\))

For the variable \(b\): \(\frac{b^{9}}{b^{5}}=b^{9 - 5}=b^{4}\)

Step 2: Combine the results

Multiply the simplified terms for \(a\) and \(b\) together: \(a^{6}\times b^{4}=a^{6}b^{4}\)

Answer:

The U.S. national debt in 2015 was approximately \(1.829\times10^{13}\) dollars.

Problem 10 (Simplify the expression \(\frac{a^{7}b^{9}}{ab^{5}}\))