24 en route vers lalgèbre. reduis les expressions algébriques suivantes en te servant des lois des exposants. ecris tes réponses en notation exponentielle en nutilisant que des exposants positifs.
a) $\frac{x^7}{x^4}$
b) $c^3 \cdot c^5$
c) $\frac{8^7}{8^5}$
d) $\frac{ab^5}{a^3b^3}$
e) $\frac{(ab^2)^3}{a^2b^2d}$
f) $\frac{x^3y^5z^4}{x^7y^2z}$
g) $(x^2y)^4 + (xy^3)^2$
h) $\left(\frac{x^3y^4}{x^2}\
ight)^2 \cdot \left(\frac{x^3y}{xy}\
ight)^4$

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Apply quotient rule of exponents ($\frac{a^m}{a^n}=a^{m - n}$)
$\frac{x^7}{x^4}=x^{7 - 4}$
## Step2: Simplify the exponent
$x^{7 - 4}=x^3$
# Answer:
$x^3$

### Part (b)
# Explanation:
## Step1: Apply product rule of exponents ($a^m\cdot a^n=a^{m + n}$)
$c^3\cdot c^5=c^{3 + 5}$
## Step2: Simplify the exponent
$c^{3 + 5}=c^8$
# Answer:
$c^8$

### Part (c)
# Explanation:
## Step1: Express 8 as $2^3$
$\frac{8^3}{2^5}=\frac{(2^3)^3}{2^5}$
## Step2: Apply power of a power rule ($(a^m)^n=a^{m\cdot n}$)
$\frac{(2^3)^3}{2^5}=\frac{2^{9}}{2^5}$
## Step3: Apply quotient rule of exponents
$2^{9 - 5}=2^4$
## Step4: Calculate $2^4$
$2^4 = 16$
# Answer:
$16$ (or $2^4$)

### Part (d)
# Explanation:
## Step1: Apply quotient rule for each variable (for $a$: $\frac{a^m}{a^n}=a^{m - n}$, for $b$: $\frac{b^m}{b^n}=b^{m - n}$)
$\frac{ab^5}{a^3b^2}=a^{1 - 3}\cdot b^{5 - 2}$
## Step2: Simplify each exponent
$a^{-2}\cdot b^{3}$
## Step3: Rewrite $a^{-2}$ as $\frac{1}{a^2}$ (optional, but if we want positive exponents) or leave as $a^{-2}b^3$. But usually, we can write with positive exponents: $\frac{b^3}{a^2}$
# Answer:
$\frac{b^3}{a^2}$ (or $a^{-2}b^3$)

### Part (e)
# Explanation:
## Step1: Apply power of a product rule ($(ab)^n=a^n b^n$)
$\frac{(ab^2)^3}{a^2b^2}=\frac{a^3(b^2)^3}{a^2b^2}$
## Step2: Apply power of a power rule
$\frac{a^3b^6}{a^2b^2}$
## Step3: Apply quotient rule for each variable
$a^{3 - 2}\cdot b^{6 - 2}$
## Step4: Simplify each exponent
$a^1\cdot b^4=ab^4$
# Answer:
$ab^4$

### Part (f)
# Explanation:
## Step1: Apply quotient rule for each variable ($x$, $y$, $z$)
$\frac{x^3y^5z^{-4}}{x^{-2}y^2z}=\ x^{3-(-2)}\cdot y^{5 - 2}\cdot z^{-4 - 1}$
## Step2: Simplify each exponent
$x^{5}\cdot y^{3}\cdot z^{-5}$
## Step3: Rewrite $z^{-5}$ as $\frac{1}{z^5}$ (or leave with negative exponent: $x^5y^3z^{-5}$)
$\frac{x^5y^3}{z^5}$
# Answer:
$\frac{x^5y^3}{z^5}$ (or $x^5y^3z^{-5}$)

### Part (g)
# Explanation:
## Step1: Apply power of a power rule to each term
$(x^2y)^4=x^{2\cdot4}y^{1\cdot4}=x^8y^4$; $(xy^3)^2=x^{1\cdot2}y^{3\cdot2}=x^2y^6$
## Step2: Apply product rule of exponents (multiply the two terms)
$x^8y^4\cdot x^2y^6=x^{8 + 2}\cdot y^{4+6}$
## Step3: Simplify each exponent
$x^{10}y^{10}$
# Answer:
$x^{10}y^{10}$

### Part (h)
# Explanation:
## Step1: Simplify the first fraction $\frac{x^3y^4}{x^2}$ using quotient rule
$\frac{x^3y^4}{x^2}=x^{3 - 2}y^4=xy^4$
## Step2: Apply power of a power rule to the first part: $(xy^4)^2$
$(xy^4)^2=x^{2}y^{8}$
## Step3: Simplify the second fraction $\frac{x^3y}{xy}$ using quotient rule
$\frac{x^3y}{xy}=x^{3 - 1}y^{1 - 1}=x^2$
## Step4: Apply power of a power rule to the second part: $(x^2)^3$
$(x^2)^3=x^{6}$
## Step5: Multiply the two results (from step 2 and step 4)
$x^{2}y^{8}\cdot x^{6}=x^{2+6}y^{8}$
## Step6: Simplify the exponent of $x$
$x^{8}y^{8}$
# Answer:
$x^{8}y^{8}$

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Apply quotient rule of exponents ($\frac{a^m}{a^n}=a^{m - n}$)

Step2: Simplify the exponent

Step1: Apply product rule of exponents ($a^m\cdot a^n=a^{m + n}$)

Step2: Simplify the exponent

Step1: Express 8 as $2^3$

Step2: Apply power of a power rule ($(a^m)^n=a^{m\cdot n}$)

Step3: Apply quotient rule of exponents

Step4: Calculate $2^4$

Answer:

Part (b)