9. national debt in 1915, the u.s. national debt was approximately 3.1×10² dollars. in 2015, the u.s. national debt was 5.9×10⁹ times more than in 1915. what was the u.s. national debt in 2015? simplify each expression. assume that no denominator equals zero. 10. $\frac{a^7b^9}{ab^3}$ 11. $\frac{c^4d^5}{c^4d^2}$ 12. $\frac{20a^4f^3g^2}{5a^3fg^3}$ 13. $\frac{-18h^4}{2h^3}$ 14. $\frac{5y^4}{xy^4}$ 15. $\frac{15k^6}{3k^5}$

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Accepted Answer

### Problem 10: Simplify $\boldsymbol{\frac{a^7b^9}{ab^3}}$
# Explanation:
## Step 1: Apply Quotient Rule for Exponents ($ \frac{x^m}{x^n} = x^{m - n} $) to $a$ terms.
For the $a$ terms: $\frac{a^7}{a^1}=a^{7 - 1}=a^6$
## Step 2: Apply Quotient Rule for Exponents to $b$ terms.
For the $b$ terms: $\frac{b^9}{b^3}=b^{9 - 3}=b^6$
## Step 3: Combine the results.
Multiply the simplified $a$ and $b$ terms: $a^6 \cdot b^6 = a^6b^6$
# Answer:
$a^6b^6$

### Problem 11: Simplify $\boldsymbol{\frac{c^4d^5}{c^4d^2}}$
# Explanation:
## Step 1: Apply Quotient Rule for Exponents to $c$ terms.
For the $c$ terms: $\frac{c^4}{c^4}=c^{4 - 4}=c^0 = 1$ (since $x^0 = 1$ for $x
eq0$)
## Step 2: Apply Quotient Rule for Exponents to $d$ terms.
For the $d$ terms: $\frac{d^5}{d^2}=d^{5 - 2}=d^3$
## Step 3: Combine the results.
Multiply the simplified $c$ and $d$ terms: $1 \cdot d^3 = d^3$
# Answer:
$d^3$

### Problem 12: Simplify $\boldsymbol{\frac{20a^4f^3g^2}{5a^2f g^3}}$
# Explanation:
## Step 1: Simplify the coefficient.
$\frac{20}{5}=4$
## Step 2: Apply Quotient Rule for Exponents to $a$ terms.
$\frac{a^4}{a^2}=a^{4 - 2}=a^2$
## Step 3: Apply Quotient Rule for Exponents to $f$ terms.
$\frac{f^3}{f^1}=f^{3 - 1}=f^2$
## Step 4: Apply Quotient Rule for Exponents to $g$ terms.
$\frac{g^2}{g^3}=g^{2 - 3}=g^{-1}=\frac{1}{g}$ (since $x^{-n}=\frac{1}{x^n}$)
## Step 5: Combine the results.
Multiply the simplified coefficient, $a$, $f$, and $g$ terms: $4 \cdot a^2 \cdot f^2 \cdot \frac{1}{g}=\frac{4a^2f^2}{g}$
# Answer:
$\frac{4a^2f^2}{g}$

### Problem 13: Simplify $\boldsymbol{\frac{-18n^4}{2n^3}}$
# Explanation:
## Step 1: Simplify the coefficient.
$\frac{-18}{2}=-9$
## Step 2: Apply Quotient Rule for Exponents to $n$ terms.
$\frac{n^4}{n^3}=n^{4 - 3}=n^1 = n$
## Step 3: Combine the results.
Multiply the simplified coefficient and $n$ term: $-9 \cdot n=-9n$
# Answer:
$-9n$

### Problem 14: Simplify $\boldsymbol{\frac{x^5y^4}{xy^4}}$
# Explanation:
## Step 1: Apply Quotient Rule for Exponents to $x$ terms.
$\frac{x^5}{x^1}=x^{5 - 1}=x^4$
## Step 2: Apply Quotient Rule for Exponents to $y$ terms.
$\frac{y^4}{y^4}=y^{4 - 4}=y^0 = 1$
## Step 3: Combine the results.
Multiply the simplified $x$ and $y$ terms: $x^4 \cdot 1 = x^4$
# Answer:
$x^4$

### Problem 15: Simplify $\boldsymbol{\frac{15x^6}{3x^5}}$
# Explanation:
## Step 1: Simplify the coefficient.
$\frac{15}{3}=5$
## Step 2: Apply Quotient Rule for Exponents to $x$ terms.
$\frac{x^6}{x^5}=x^{6 - 5}=x^1 = x$
## Step 3: Combine the results.
Multiply the simplified coefficient and $x$ term: $5 \cdot x = 5x$
# Answer:
$5x$

### Problem 9: National Debt Calculation
(Assuming the 1915 debt is $3.1\times 10^9$ and 2015 is $5.9\times 10^9$ times more? Wait, likely a typo, maybe 2015 debt is $5.9\times 10^9$ times the 1915 debt. 1915 debt: $3.1\times 10^9$ dollars. Then 2015 debt is $3.1\times 10^9\times5.9\times 10^9$)
# Explanation:
## Step 1: Multiply the coefficients.
$3.1\times5.9 = 18.29$
## Step 2: Multiply the powers of 10 (using $10^m\times10^n = 10^{m + n}$)
$10^9\times10^9 = 10^{9 + 9}=10^{18}$
## Step 3: Combine the results.
$18.29\times10^{18}$. In scientific notation, we adjust to $1.829\times 10^{19}$ (since $18.29 = 1.829\times 10^1$, so $1.829\times 10^1\times10^{18}=1.829\times 10^{19}$)
# Answer:
If 1915 debt is $3.1\times 10^9$ and 2015 is $5.9\times 10^9$ times that, then 2015 debt is $\boldsymbol{1.829\times 10^{19}}$ dollars (or approximately $1.8\times 10^{19}$ as a rough estimate).

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Question

Explanation:

Problem 10: Simplify \(\boldsymbol{\frac{a^7b^9}{ab^3}}\)

Step 1: Apply Quotient Rule for Exponents (\( \frac{x^m}{x^n} = x^{m - n} \)) to \(a\) terms.

Step 2: Apply Quotient Rule for Exponents to \(b\) terms.

Step 3: Combine the results.

Step 1: Apply Quotient Rule for Exponents to \(c\) terms.

Step 2: Apply Quotient Rule for Exponents to \(d\) terms.

Step 3: Combine the results.

Step 1: Simplify the coefficient.

Step 2: Apply Quotient Rule for Exponents to \(a\) terms.

Step 3: Apply Quotient Rule for Exponents to \(f\) terms.

Step 4: Apply Quotient Rule for Exponents to \(g\) terms.

Step 5: Combine the results.

Answer:

Problem 11: Simplify \(\boldsymbol{\frac{c^4d^5}{c^4d^2}}\)